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1. E Ex class 158 APPENDIX A MATH SYMBOL TOKENS FOR SET THEORY Token Symbol Token Symbol Token Symbol n n R R q q R R J J lt R lt R K K CC C L L RR R M M 0 0 N N 1 1 W W i i X X T Y Y Z Z Xs N N lt lt N w N N gt gt lt N lt N lt _ lt pQ pQ gt _ ee po PQ NN N Q o ZZ Z Q Q QQ Q 1Q lo 2 2 Q o 3 3 Q Q 4 4 Q xQ 5 5 lt Q lt Q 6 6 P P T 7 1P lp 8 8 P p 9 9 P P lt P lt P pR TpR pR PR R NR R R OR OR 1R Ir 1R lr Appendix B Compressed Proofs The proofs in the set mm set theory database are stored in compressed format for efficiency Normally you needn t concern yourself with the com pressed format since you can display it with the usual proof display tools in the Metamath program show proof or convert it to the normal RPN proof format described in the Section 4 3 with save proof label normal However for sake of completeness we describe the format here and show how it maps to the normal RPN proof format A compressed proof located between and keywords consists of a left parenthesis a sequence of statement labels a right parenthesis and a sequence of upper case letters A through Z with optional white space b
2. Reg D T F x y gt Ja a y AYz z E xz gt 72 y Inf D T 2 dz y E zAz 2 gt AC D T F daVyV2z y E zAz E w gt Jwvy dw y E zAz E w A y whwex oy w H da ye x AVy y Ex C 3 8 Example 6 Class Notation in Set Theory A powerful device that makes set theory easier and that we have been using all along in our informal expository language is class abstraction no tation The definitions we introduce are rigorously justified as conservative by Takeuti and Zaring 59 or Quine 48 The key idea is to introduce the notation x which means the class of all x such that for abstraction classes and introduce meta variables that range over them An abstraction class may or may not be a set depending on whether it exists as a set A class that does not exist is called a proper class To illustrate the use of abstraction classes we will provide some exam ples of definitions that make use of them the empty set class union and unordered pair Many other such definitions can be found in the Metamath set theory database set mm We extend CN of Example 5 with new symbols class 2 U where the inner braces and last comma are constant symbols As before our dual use of some mathematical symbols for both our expository language and as primitives of the formal system should be clear from context We extend VR of Example 5
3. 00 145 5 6 13 save new_proof Command 146 5 7 Creating BIEX Output 2 ee 147 5 7 1 open tex Command 00 147 5 7 2 close tex Command 147 5 8 Creating HTML Output 2 0 0 000020 148 5 8 1 The Typesetting Comment t 148 5 8 2 write theoremlist Command 150 5 8 3 write bibliography Command 151 5 8 4 write recent additions Command 151 Pui pan aiii Ae aah ii io doe aed do ia e a 151 5 9 1 tools Command 00 151 5 9 2 help Command in tools 152 152 ee ee eee 153 155 159 161 paun edee Sle Goa i ee ae aR ae ate A 161 ea Gee SY E ee ate cee ee 162 C 2 1 Preliminaries 2 002 2 eee 162 neereere 162 itution esor ee a a 163 C 2 4 Statements 0 0 ee ee ee 163 hy Macnee ania Geko bite ae Gin Sen eee ee 165 Be Poe Metta b fie te ee a 166 C 4 Metamath as a Formal System D The MIU System 177 Bibliography 181 CONTENTS vii 187 vill CONTENTS Preface Overview Metamath is a computer language and an associated computer program for archiving verifying and studying mathematical proofs at a very de tailed level The Metamath language incorporates no mathematics per se but treats all mathematical statements as mere sequences of symbols You provide Metamath with certain special sequences axioms that tell it what rules of
4. 2 000 133 5 2 11 more Command 200 133 Pts cea heh ae 133 5 2 13 Size Limitations in Metamath 134 fe Gd te Se ee ee ee Be a oe 134 5 3 1 read Command 200 134 5 3 2 write source Command 134 jerk 549 vance teh Bost es we 135 5 4 1 show settings Command 135 5 4 2 show memory Command 135 5 4 3 show labels Command 135 5 4 4 show statement Command 135 5 4 5 search Command 136 te rae fees aes aes eee ae ie 136 5 5 1 show proof Command 136 5 5 2 show usage Command 0 137 5 5 3 show trace back Command 138 5 5 4 verify proof Command 138 5 5 5 save proof Command 138 a Sy HVE Soh a EL th e Se Se ae Re 139 5 6 1 prove Command 00 141 5 6 2 set unification timeout Command 141 5 6 3 set empty_substitution Command 142 5 6 4 set search limit Command 142 5 6 5 show new proof Command 142 5 6 6 assign Command 00 143 5 6 7 match mmand s gt sss s sa aeaa aa a 143 5 6 8 let Command aaa aaa a 143 5 6 9 unify Command oaa a 144 5 6 10 initialize Command vi CONTENTS 5 6 11 delete Command 145 5 6 12 improve Command
5. 3 3 4 Set Theory In order to make the axioms of set theory a little more compact there are several definitions from logic that we make use of implicitly namely logical AND logical equivalence and there exists yAw stands for y gt y py stands for p gt A Y gt day stands for Var Ay In addition the axioms of set theory require that all variables be dis tinct f thus we also assume d xyzw Axiom of Extensionality ax ext a F Va reyouez gt y z Axiom of Replacement ax rep a H Yw3yYz Yyy gt z y gt J3yYz zEyJw wETA Vyy Axiom of Union ax un a FdaVy da yerArez ryen Axiom of Power Sets 4Set theory axioms can be devised so that no variables are required to be distinct provided we replace ax 16 with an axiom stating that at least two things exist thus making ax 17 the only other axiom requiring the d statement These axioms are uncon ventional and are not presented here but they can be found on the http metamath org web site See also the Comment on p 105 66 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED ax pow a FidarVy Va reyorez ryer Axiom of Regularity ax reg a da reyoda reyAVz zEer772z y Axiom of Infinity ax inf a Fda yerAVy yexrdz yezAzez Axiom of Choice ax ac a FdaVyV2 yezAzew osruVy dw yezAzEew A yeEwNwer oy w 3 3 5 That s It There you have it the axio
6. Given any theorem with UU in the middle it remains a theorem if the UU is deleted IV a l xy Now we prove the theorem MUIIU You may be interested in comparing this proof with that of Hofstadter pp 35 36 theoreml p MUIIU we wM wU wl we wI wU we wU wI wU we wM we wI wU we wM wI wI wI we wI wI we wI ax II II I_ III II IV The show proof essential lemmon renumber command yields the following display It is very similar to the one in 24 pp 35 36 1 ax a MI 2 1 II a MII 3 2 II a MIIII 43 1_ a MIIIIU 5 4 III a MU IU 65 II a MUIUUIU 76 IV a MUIIU We note that Hofstadter s MU puzzle which asks whether MU is a theorem of the MIU system cannot be answered using the system above because the MU puzzle is a question about the system To prove the answer to the MU puzzle a much more elaborate system is needed namely one that models the MIU system within set theory Incidentally the answer to the MU puzzle is no 180 APPENDIX D THE MIU SYSTEM Bibliography Donald J Albers and G L Alexanderson editors Mathematical People Contemporary Books Inc Chicago 1985 QA28 M37 Alan Ross Anderson and Nuel D Belnap Entailment volume 1 Prince ton University Press Princeton 1975 QA9 A634 1975 v 1 John D Barrow Theories of Everything The Quest for Ultimate Ex planation Oxford University Press Oxford 1991 Q175 B225 H B
7. Define the union of two classes Definition 5 6 of Takeuti and Zaring p 16 df un a H AUB xz xeAvzeB df un a Au B x1 1Cxe A xe B amp Define the intersection of two classes Definition 5 6 of Takeuti and Zaring p 16 df in a k ANB a t E AArceEB df in a Ai7iB xIlCxe A xe B Define class difference Definition 5 12 of Takeuti and Zaring p 20 Several notations are used in the literature we chose the convention instead of a minus sign to reserve the latter for later use in e g arithmetic df dif a A B 2 rtEe AA7zrEB df dif a A B x1 1Cxe A xe B Define the empty or null set Compare Definition 5 14 of Takeuti and Zaring p 20 df nul a F V V df nul a l J V V amp Define power class Definition 5 10 of Takeuti and Zaring p 17 but we also let it apply to proper classes Note that P is the symbol for calligraphic P the tilde suggesting curly see Appendix A df pw a Kk PA a xCA df pw a P gt A x x _A Define the singleton of a class Definition 7 1 of Quine p 48 It is well defined for proper classes although it is not very meaningful in this case where it evaluates to the empty set df sn a A a x A df sn a A f xIlx A Define an unordered pair of classes Definition 7 1 of Quine p 48 df pr a F A B A U B df pr a f A B C A u B F Define an unordered triple of classes Definiti
8. HTML file normally called mmrecent html and updates it with the descriptions of the most recently added theorems to the database The file is updated between the HTML comment lines lt START gt and lt END gt The original input file is renamed to filename 1 Optional command qualifier limit number This qualifier specifies the number of most recent the orems to write to the output file The default is 100 5 9 Text File Utilities 5 9 1 tools Command Syntax tools This command invokes an easy to use general purpose utility for manip ulating the contents of ASCII text files Upon typing tools the command line prompt will change to TOOLS gt until you type exit The tools com mands can be used to perform simple global edits on an input output file such making a character string substitution on each line adding a string to each line and so on A typical use of this utility is to build a submit input file to perform a common operation on a list of statements obtained from show label or show usage The actions of most of the tools commands can also be performed with equivalent and more powerful Unix shell commands and some users may find those more efficient But for Windows users or users not comfortable with Unix tools provides an easy to learn alternative that is adequate for most of the script building tasks needed to use the Metamath program ef fectively 152 CHAPTER 5 THE METAMATH PROGRAM
9. The file name may not contain a or white space The file must exist The case sensitivity of its name follows the conventions of the operating system The contents of the file replace the inclusion command Included files may include other files Only the first reference to a given file is included any later references to the same file whether in the top level file or in included files cause the inclusion command to be ignored treated like white space A file self reference is ignored as is any reference to the top level file Like all tokens the and keywords must be surrounded by white space 4 1 3 Basic Syntax After preprocessing a database will consist of a sequence of statements It may contain only the statement types defined below These are the scop ing statements and along with the c v f e d a and p statements A scoping statement consists only of its keyword or begins a block and a matching ends the block Every must have a matching Defining it recursively we say a block contains a sequence of zero or more tokens other than and and possibly other blocks There is an outermost block not bracketed by the end of the outermost block is the end of the database A v or c statement consists of the keyword token v or c respec tively followed by one or more math symbols followed by These state ments declare the math symbols to be variables or co
10. to mean a subset of and belongs to or is contained in to mean is an element of What we call a formal description here unlike earlier is actually an informal description in the ordinary language of mathematicians However we provide sufficient detail so that a mathematician could easily formalize it even in the language of Metamath itself if desired To understand the logic examples at the end of this appendix familiarity with an introductory book on mathematical logic would be helpful 10 p 3 25 161 162 APPENDIX C METAMATH S FORMAL SYSTEM C 2 The Formal Description C 2 1 Preliminarie lt By w we denote the set of all natural numbers non negative integers Each natural number n is identified with the set of all smaller numbers n m m lt n The formula m lt n is thus equivalent to the condition m n and m n w In particular 0 is the number zero and at the same time the empty set 1 0 2 0 1 etc PA denotes the set of all functions on B to A i e with domain B and range included in A The members of A are what are called simple infinite sequences with all terms in A In case n w the members of A are referred to as finite n termed sequences again with terms in A The consecutive terms function values of a finite or infinite sequence f are denoted by fo fi fn Every finite sequence f Uncu A uniquely determines the number n such that f
11. 5 5 4 verify proof Command Syntax verify proof label match syntax_only This command verifies the proofs of the specified statements label match may contain wild card characters to verify more than one proof for example abc def will match all labels containing abc and ending with def verify proof will verify all proofs in the database Optional command qualifier syntax_only This qualifier will perform a check of syntax and RPN stack violations only It will not verify that the proof is correct This qualifier is useful for quickly determining which proofs are incomplete i e are under development and have s in them Note read followed by verify proof will ensure the database is free from errors in Metamath language but will not check the markup notation in comments for that see Section 5 5 5 save proof Command Syntax save proof label normal compressed The save proof command will reformat a proof in one of two formats and replace the existing proof in the source buffer It is useful for converting between proof formats Note that a proof will not be permanently saved until awrite source command is issued Optional command qualifiers normal The proof is saved in the normal format i e as a sequence of labels which is the defined format of the basic Metamath language This is the default format that is used if a qualifier is omitted 5 6 CREATING PROOFS 139 compressed The proof is save
12. A yE zaA x lt x y gt e F gt x 2z Define an onto function Definition 6 15 4 of Takeuti and Zaring p 27 df fo a H F A Bo F Fn AdAran F B onto df fo a F A onto gt B lt gt F Fn A ran F 76 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED B Define a one to one onto function Compare Definition 6 15 6 of Takeuti and Zaring p 27 1 z df f1o a H F A Be F A BAF A B df fio a F A 1 1 onto gt B lt gt F A 1 1 gt B F A onto gt B Define the value of a function This definition applies to any class and evaluates to the empty set when it is not meaningful Note that F A means the same thing as the more familiar F A notation for a function s value at A The F A notation is common in formal set theory df fv a F F A Uf al F A 2 df fv a F A U x CF A x 6 Define the result of an operation Here F is an operation on two values such as for real numbers This is defined for proper classes A and B even though not meaningful in that case However the definition can be meaningful when F is a proper class df opr a H A F B F A B df opr a AFB CF lt A B gt 3 5 Tricks of the Trade In the set mm database our goal was usually to conform to modern notation However in some cases the relationship to standard textbook language may be obscured by several unconventional devices we used to sim
13. This might be true if the proof were to be expressed directly in terms of the axioms of logic and set theory but it is usually not true if we allow ourselves a hierarchy of definitions and theorems to build upon using a notation that allows us to introduce new symbols definitions and theorems in a precisely specified way Even in formal logic formal proofs that are considered complete still contain hidden or implicit information For example a proof is usually defined as a sequence of wits P each of which is an axiom or follows from a rule applied to previous wffs in the sequence The implicit part of the proof is the algorithm by which a sequence of symbols is verified to be a valid wff given the definition of a wff The algorithm in this case is rather simple but for a computer to verify the proof it must have the algorithm built into its verification program If one deals exclusively with axioms and elementary wits it is straightforward to implement such an algorithm But as more and more definitions are added to the theory in order to make the expression of wffs more compact the algorithm becomes more and more complicated A computer program that implements the algorithm becomes larger and harder to understand as each definition is introduced and thus more prone to bugs The larger the program the more suspicious the mathematician may be about the validity of its algorithms This is especially true because computer programs
14. wf Y wif y ie those expressions of length 2 whose first member is wff and whose second member belongs to VR Then T consists of the axiomatic statements that are the reducts of following pre statements wE p gt w wif y F y gt b gt 9 F gt gt x gt gt y gt e gt x 4For convenience we let T be an infinite set the definition of a statement permits this in principle Since a Metamath source file has a finite size in practice we must of course use appropriate finite subsets of this T specifically ones containing at least the mandatory variable type hypotheses Similarly in the source file we introduce new variables as required with the understanding that a potentially infinite number of them are available T T T a ae T C 3 EXAMPLES OF FORMAL SYSTEMS 167 2 T 2 Y p E Y 2 T F y gt E et E Y For example the reduct of T wf p gt w is D wE p wit Y wE p gt w which is the first axiomatic statement We call the members of VR wff variables or in the context of first order logic which we will describe shortly wff metavariables Note that the sym bols w etc denote actual specific members of VR they are not metavari ables of our expository language which we denote with a 6 etc but are instead meta constant symbols members of SM from the
15. 30 40 50 60 70 80 90 100 All proofs in the database were verified in 2 84 s No errors were reported so every proof is correct You need to know the names labels of theorems before you can look at them Often just examining the database file s with a text editor is the best approach In set mm there are many detailed comments especially near the beginning that can help guide you The search command in the Metamath program is also handy The comments qualifier will list the statements whose associated comment the one immediately before it contain a string you give it For example if you are studying Enderton s Elements of Set Theory I6 you may want to see the references to it in the database The search string enderton is not case sensitive This will not show you all the database theorems that are in Enderton s book because there is usually only one citation for a given theorem which may appear in several textbooks MM gt search enderton comments 3332 df ral a niversal quantification Enderton p 22 3333 df rex a stential quantification Enderton p 22 4828 df tp a of classes Definition of Enderton p 19 5200 ssuniss p nd union Exercise 5 of Enderton p 26 5217 opeluu p air belongs Lemma 3D of Enderton p 41 etc Or you may want to see what theorems have something to do with con junction logical AND The quotes around the search string are optional when the
16. A n is called the length of f and is denoted by f a is the sequence f with f 1 and fo a a b is the sequence f with f 2 fo a fi b etc Given two finite sequences f and g we denote by f gt g their concatenation i e the finite sequence h determined by the conditions h f lgl hn fin for n lt f hy flan Jn for n lt g C 2 2 Constants Variables and Expressions A formal system has a set of symbols denoted by SM A precise set theo retical definition of this set is unimportant a symbol could be considered a primitive or atomic element if we wish We assume this set is divided into two disjoint subsets a set CN of constants and a set VR of variables CN and VR are each assumed to consist of countably many symbols which may be arranged in finite or simple infinite sequences co c1 and vo V1 respectively without repeating terms We will represent arbitrary symbols by metavariables a 8 etc Comment The variables vo v1 of our formal system correspond to what are usually considered metavariables in descriptions of specific formal systems in the literature Typically when describing a specific formal system a book will postulate a set of primitive objects called variables then proceed to describe their properties using metavariables that range over them never mentioning again the actual variables themselves Our formal system does not mention these primitive variable o
17. Define the restriction of a class Definition 6 6 1 of Takeuti and Zaring p 24 df res a F A B AN BxV df res a CA 1 B CAi7i BX V amp Define the image of a class Definition 6 6 2 of Takeuti and Zaring p 24 df ima a A B ran A B df ima a A B ran CA B Define the composition of two classes Definition 6 6 3 of Takeuti and Zaring p 24 df co a AoB 2 y 4dz 2 z EBA z y EA df co a Ao B lt x y gt E z lt x 2 gt e B lt z y gt e A S Define a relation Definition 6 4 1 of Takeuti and Zaring p 23 df rel a Rel AG AC VxV df rel a Rel A lt gt A _ VX V amp Define a function Definition 6 4 4 of Takeuti and Zaring p 24 df fun a Fun A Rel AAVadzVy 2 y e A gt y z df fun a Fun A lt gt ReL A A XE zA y CK x y gt e A gt y z Define a function with domain Definition 6 15 1 of Takeuti and Zaring p 27 df fn a H A Fn BY Fun A dom A B df fn a A Fn B lt gt Fun A dom A B Define a function with domain and co domain Definition 6 15 3 of Takeuti and Zaring p 27 df f a H F A B amp F Fn A ran FCB df f a F A gt B lt gt F Fn A ran F _B Define a one to one function Compare Definition 6 15 5 of Takeuti and Zaring p 27 df f1 a H F A 14 Bo F ASBAVyAzVa 2 y For z df f1 a F A 1 1 gt B lt gt F A gt B
18. This is the default and it is redundant to specify it all the proof tree is not trimmed of all f hypotheses before being displayed essential and all are mutually exclusive from_step step The display starts at the specified step If this qualifier is omitted the display starts at the first step to_step step The display ends at the specified step If this qualifier is omitted the display ends at the last step tree_depth number Only steps at less than the specified proof tree depth are displayed Sometimes useful for obtaining an overview of the proof reverse The steps are displayed in reverse order renumber When used with essential the steps are renumbered to correspond only to the essential steps tex The proof is converted to TEX and stored in the file opened with open tex See Section 5 7 or help tex in the program lemmon The proof is displayed in a non indented format known as Lemmon style with explicit previous step number references If this qualifier is omitted steps are indented in a tree format start_column number Overrides the default column at which the formula display starts in a Lemmon style display May be used only in conjunction with lemmon normal The proof is displayed in normal format suitable for inclusion in a Metamath source file May not be used with any other qualifier compressed The proof is displayed in compressed format suitable for inclusion in a Metamath so
19. al though nothing along those lines has been done so far For constructing proofs the Metamath program has a Proof Assistant which helps you fill in some of a proof step s details shows you what choices you have at any step and verifies the proof as you build it but you are still expected to provide the proof Like most computer languages the Metamath language uses the standard ASCII characters on a computer keyboard so it cannot directly represent many of the special symbols that mathematicians use A useful feature of the Metamath program is its ability to convert its notation into the BT X typesetting language This feature lets you convert the ASCII tokens you ve defined into standard mathematical symbols so you end up with symbols and formulas you are familiar with instead of somewhat cryptic ASCII rep resentations of them Metamath is probably conceptually different from anything you ve seen before and some aspects may take some getting used to This book will help you decide whether Metamath suits your specific needs Setting Your Expectations It is important for you to understand what Metamath is and is not As men tioned Metamath is not an automated theorem prover but rather a proof verifier Developing a database can be tedious hard work especially if you want to make the proofs as short as possible but it becomes easier as you build up a collection of useful theorems The purpose of Metamath is sim ply to doc
20. not strengthen the language Theorem bii shows the substitution needed to eliminate the symbol Definition df cleq extends the usage of the equality symbol to include classes in set theory The reason it is potentially problematic is that it can lead to statements which do not follow from logic alone but presuppose the Axiom of Extensionality so we include this axiom as a hypothesis for the definition We could have made df cleq directly eliminable by introducing a new equality symbol but have chosen not to do so in keeping with standard textbook practice Definitions such as df cleq that extend the meaning of existing symbols must be introduced carefully so that they do not lead to contradictions Definition df clel also extends the meaning of an existing 5Here we mean the elimination that a human might do in his or her head To eliminate them as part of a Metamath proof we would invoke one of a number of theorems that deal with transitivity of equivalence or equality there are many such examples in the proofs in set mm 68 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED symbol while it doesn t strengthen the language like df cleq this is not obvious and it must also be subject to the same scrutiny Exercise Study how the wff x w meaning zx is a natural number could be expanded in terms of primitive symbols starting with the defini tions df clel on p 71 and df om on p 74 and working your way back Don t b
21. on the other hand is tt since thi occurs outside of the block containing min and maj Note that and do not affect the scope of assertions a and p Assertions are always available to be referenced by any later proof in the source file Each provable assertion p statement has two parts The first part is the assertion itself which is a sequence of math symbol tokens placed between the p token and a token The second part is a proof which is a list of label tokens placed between the token and the token that ends 40 CHAPTER 2 USING THE METAMATH PROGRAM the statement The proof acts as a series of instructions to the Metamath program telling it how to build up the sequence of math symbols contained in assertion part of the p statement making use of the hypotheses of the p statement and previous assertions The construction takes place according to precise rules If the list of labels in the proof causes these rules to be violated or if the final sequence that results does not match the assertion the Metamath program will notify you with an error message If you are familiar with reverse Polish notation RPN which is some times used on pocket calculators here in a nutshell is how a proof works Each hypothesis label in the proof is pushed onto the RPN stack as it is encountered Each assertion label pops off the stack as many entries as the referenced assertion has mandatory hypotheses Variable substitut
22. please specify SETTINGS STATEMENT or SOURCE 46 CHAPTER 2 USING THE METAMATH PROGRAM If you don t what know argument to use as part of a command type a at the argument position Metamath will tell you what it expected there MM gt show Expected SETTINGS LABELS STATEMENT SOURCE PROOF MEMORY TRACE_BACK or USAGE Finally you may type just the first word or words of a command followed by return Metamath will prompt you for the remaining part of the com mand showing you the choices at each step For example instead of typing show statement thi full you could interact in the following manner MM gt show SETTINGS LABELS STATEMENT SOURCE PROOF MEMORY TRACE_BACK or USAGE lt SETTINGS gt st What is the statement label lt thi gt or nothing lt nothing gt TEX COMMENT_ONLY or FULL lt TEX gt f or nothing lt nothing gt 19 thi p l t t After each in this mode you must give Metamath the information it requests Sometimes Metamath gives you a list of choices with the default choice indicated by brackets lt gt Pressing return after the will select the default choice Answering anything else will override the default Note that the in command qualifiers is considered a separate token by the parser and this is why it is asked for separately 2 4 Your First Proof Proofs are developed with the aid of the Proof Assistant We will now show you how the proof of theorem thi w
23. qualifiers see below This command provides information about a statement Only statements that have labels f e a and p may be specified If label contains wildcard characters all matching statements will be displayed in the order they occur in the database Optional qualifiers only one qualifier at a time is allowed 136 CHAPTER 5 THE METAMATH PROGRAM comment This qualifier includes the comment that immediately pre cedes the statement full Show complete information about each statement and show all statements matching label including e and f statements tex This qualifier will write the statement information to the ATRxX file previously opened with open tex See Section 5 7 simple_tex The same as tex except that TX macros are not used for formatting equations allowing easier manual edits of the output for slide presentations etc html alt_html brief_html brief_alt_html These qualifiers invoke a special mode of show statement that creates a web page for the statement They may not be used with any other qualifier See Section 5 8 or help html in the program 5 4 5 search Command Syntax search label match symbol match all comments This command searches all a and p statements matching label match for occurrences of symbol match A in label match matches any label char acter A in symbol match matches any sequence of symbols The symbols in symbol match must be se
24. so you don t need quotes around a Unix file name Thus bash metamath tmp set mm and bash metamath read tmp set mm are equivalent 5 2 Controlling Metamath The Metamath program was first developed on a VAX VMS system and some aspects of its command line behavior reflect this heritage Hopefully you will find it reasonably user friendly once you get used to it Each command line is a sequence of English like words separated by spaces as in show settings Command words are not case sensitive and only as many letters are needed as are necessary to eliminate ambiguity for example sh se would work for the command show settings In some cases arguments such as file names statement labels or symbol names are required these are case sensitive although file names may not be on some operating systems A command line is entered by typing it in then pressing the return enter key To find out what commands are available type at the MM gt prompt To find out the choices at any point in a command press return and you will be prompted for them The default choice the one selected if you just press return is shown in brackets lt gt You may also type in place of a command word to force Metamath to tell you what the choices are The method won t work though if a 5 2 CONTROLLING METAMATH 131 non keyword argument such as a file name is expected at that point because the program will think that is the
25. terms of the quadruple are called distinct variable restrictions variable type hypotheses logical hypotheses and the assertion respectively We denote by Tm mandatory variable type hypotheses the subset of T such that V Tm V H U A We denote by Dy a 8 DHa b C V Tm the mandatory distinct variable restrictions of the pre statement The set of mandatory hypotheses is Tm U H We call the quadruple Dm Tm H A the reduct of the pre statement D T H A A statement is the reduct of some pre statement A statement is therefore a special kind of pre statement in particular a statement is the reduct of itself Comment T is a set of expressions each of length 2 that associate a set of constants variable types with a set of variables The condition V HU A C V T means that each variable occurring in a statement s logical hypotheses or assertion must have an associated variable type hypothesis or type declaration in analogy to a computer programming language where a variable must be declared to be say a string or an integer The requirement that Ve f T e1 f for e f means that each variable must be associated with a unique constant designating its variable type e g a variable might be a wff or a set but not both Distinct variable restrictions are used to specify what variable substi tutions are permissible to make for the statement to remain valid For example in the theorem sch
26. you will find that Metamath is not a practical tool for manipulating numbers Even proving that 2 2 4 if you start with set theory can be quite complex Several commercial mathematics pack ages are quite good at arithmetic algebra and calculus and as practical tools they are invaluable But they have no notion of proof and cannot understand statements starting with there exists such and such Software packages such as Mathematica do not concern themselves with proofs but instead work directly with known results These pack ages primarily emphasize heuristic rules such as the substitution of equals for equals to achieve simpler expressions or expressions in a different form Starting with a rich collection of built in rules and algorithms users can add to the collection by means of a powerful programming language However results such as say the existence of a certain abstract object without dis playing the actual object cannot be expressed directly in their languages The idea of a proof from a small set of axioms is absent Instead this soft ware simply assumes that each fact or rule you add to the built in collection of algorithms is valid One way to view the software is as a large collection 1 3 THE USE OF COMPUTERS IN MATHEMATICS 25 of axioms from which the software with certain goals attempts to derive new theorems for example equating a complex expression with a simpler equivalent But the terms theorem
27. 119 122 1978 QA N914 Willard Van Orman Quine Set Theory and Its Logic The Belknap Press of Harvard University Press Cambridge Massachusetts revised edition 1969 QA248 Q7 1969 J A Robinson A machine oriented logic based on the resolution prin ciple Journal of the Association for Computing Machinery 12 23 41 1965 T Thacher Robinson Independence of two nice sets of axioms for the propositional calculus The Journal of Symbolic Logic 33 265 270 1968 QA J87 BIBLIOGRAPHY 185 51 52 53 54 55 56 57 58 59 60 61 62 63 Rudy Rucker Infinity and the Mind The Science and Philosophy of the Infinite Bantam Books Inc New York 1982 QA9 R79 1982 Bertrand Russell Recent work on the principles of mathematics In ternational Monthly 4 84 1901 Bertrand Russell Mysticism and Logic and Other Essays Barnes amp Noble Books Totowa New Jersey 1981 B1649 R963 M9 1981 Joseph R Shoenfield Mathematical Logic Addison Wesley Publishing Company Inc Reading Massachusetts 1967 QA9 S52 Daniel Solow How to Read and Do Proofs An Introduction to Math ematical Thought Process John Wiley amp Sons New York 1982 QA9 S577 Harold M Stark An Introduction to Number Theory Markham Pub lishing Company Chicago 1970 QA241 S72 1978 E R Swart The philosophical implications of the four color prob lem American Mathematical Monthly 8
28. 3 4 A HIERARCHY OF DEFINITIONS 71 expression being defined and does not buy us anything in terms of brevity The reason we introduce this definition is because it fits in neatly with the extension of the connective provided by df clel df clab a F re y y ol a y y df clab a l xe y ph lt gt x y ph 6 Define the equality connective between classes See Quine or Chapter 4 of Takeuti and Zaring for its justification and methods for eliminating it This is an example of a somewhat dangerous definition because it extends the use of the existing equality symbol rather than introducing a new symbol allowing us to make statements in the original language that may not be true For example it permits us to deduce y z 4 Va a y x z which is not a theorem of logic but rather presupposes the Axiom of Exten sionality which we include as a hypothesis so that we can know when this axiom is assumed in a proof with the show trace_back command We could avoid the danger by introducing another symbol say in place of this would also have the advantage of making elimination of the defini tion straightforward and would eliminate the need for Extensionality as a hypothesis We would then also have the advantage of being able to identify exactly where Extensionality truly comes into play One of our theorems would be z y x y by invoking Extensionality However in keeping with standard practice we retain the da
29. 5 9 2 help Command in tools Syntax help The help command lists the commands available in the tools utility along with a brief description Each command in turn has its own help such as help add As with Metamath s MM gt prompt a complete command can be entered at once or just the command word can be typed causing to program to prompt for each argument Line by line editing commands add Add a specified string to each line in a file clean Trim spaces and tabs on each line in a file convert characters delete Delete a section of each line in a file insert Insert a string at a specified column in each line of a file substitute Make a simple substitution on each line of the file swap Swap the two halves of each line in a file Other file processing commands break Break up parse a file into a list of tokens one per line build Build a file with multiple tokens per line from a list count Count the occurrences in a file of a specified string number Create a list of numbers parallel Put two files in parallel reverse Reverse the order of the lines in a file right Right justify lines in a file useful before sorting numbers sort Sort the lines in a file with key starting at specified string match Extract lines containing or not a specified string unduplicate Eliminate duplicate occurrences of lines in a file duplicate Extract first occurrence of any line occurring more than once in a
30. Any statement of these 7 types located in the outermost block will remain active through the end of the database and thus are effectively global statements All c statements must be placed in the outermost block Since they are therefore always global they could be considered as belonging to both of the above categories The scope of a statement is the set of statements that recognize it as active The concept of active is also defined for math symbols Math symbols constants and variables become active in the c and v statements that declare them A variable becomes inactive when its declaration statement becomes inactive Because all c statements must be in the outermost block a constant will never become inactive after it is declared Redeclaration of Math Symbols A variable may not be declared a second time while it is active but it may be declared again after it becomes inactive This provides a convenient way to introduce local variables i e temporary variables for use in the frame of an assertion or in a proof without keeping them around forever A previously declared variable may not be redeclared as a constant A constant may not be redeclared And as mentioned above constants must be declared in the outermost block The reason variables may have limited scope but not constants is that an assertion a or p statement remains available for use in proofs through the end of the database Variables in an
31. Consider the theorem There exist irrational numbers a and b such that a is rational An intuitionist would reject the following proof If fae is rational we are done Otherwise let a A and b v2 Then a 2 which is rational 28The axiom of choice asserts that given any collection of pairwise disjoint nonempty sets there exists a set that has exactly one element in common with each set of the collection It is used to prove many important theorems in standard mathematics Some philosophers object to it because it asserts the existence of a set without specifying what the set contains p 154 In one foundation for mathematics due to Quine that has 18 CHAPTER 1 INTRODUCTION The simplicity of the Metamath language lets the algorithm computer program that verifies the validity of a Metamath proof to be straightforward and robust You can have confidence that the theorems it verifies really can be derived from your axioms 1 1 6 Rigor Rigor became a goal with the Greeks But the efforts to pursue rigor to the utmost have led to an impasse in which there is no longer any agreement on what it really means Mathematics remains alive and vital but only on a pragmatic basis Morris Kine Kline refers to a much deeper kind of rigor than that which we will discuss in this section Gd del s incompleteness theorem showed that it is impossible to achieve absolute rigor in standard mathematics because we can never prove that
32. E would be a variable and the other four symbols constants In principle a Metamath database could be constructed to work with almost any unambiguous English language math ematical statement but as a practical matter the definitions needed to pro vide for all possible syntax variations would be cumbersome and distracting and possibly have subtle pitfalls accidentally built in We generally rec ommend that you express mathematical statements with compact standard mathematical symbols whenever possible and put their English language descriptions in comments Axioms and definitions a statements are the only places where Metamath will not detect an error and doing this will help reduce the number of definitions needed You are free to use any tokens you like for math symbols Appendix A recommends token names to use for symbols in set theory and we suggest you adopt these in order to be able to include the set mm set theory database in your database For printouts you can convert the tokens in a database to standard mathematical symbols with the TEX typesetting program The Metamath command open tex filename produces output that can be read by ATEX The correspondence between tokens and the actual symbols is made by latexdef statements inside of a special database comment tagged with t You can edit this comment to change the definitions or add new ones Appendix A describes how to do this in more detail Label Tokens Label tokens are
33. Format 4 1 Specification of the Metamath Language 4 1 1 Preliminaries 4 1 2 Preprocessing 4 1 3 Basic Syntax ee eee ae eee ee stands Be Be Ae te ot ae See Be de he ek 4 2 1 User Defined Tokens 4 2 2 Constants and Variables 4 2 3 The c and v Declaration Statements 4 2 4 The d Statement 4 2 5 The f and e Statements 4 2 6 Assertions a and p Statements 4 2 8 Scoping Statements and 4 2 7 Frames 02004 CONTENTS ehhh aA 83 webb Ga 96 4 4 3 Including Other Files in a Metamath Source File 122 4 4 4 Compressed Proof Format CONTENTS v 4 4 5 Specifying Unknown Proofs or Subproofs 124 4 5 Appendix Axioms vs Definitions ooo aaa 125 5 The Metamath Program 129 5 1 Invoking Metamath oaa 129 5 2 Controlling Metamath aaa 130 5 2 1 exit Command aoaaa a 131 5 2 2 open log Command gt se rasei aea paa 131 5 2 3 close log Command 0 132 5 2 4 submit Command aaa 132 5 2 5 erase Command saaa 132 5 2 6 set echo Command 132 5 2 7 set scroll Command 132 5 2 8 set width Command 132 5 2 9 set height Command 0 133 5 2 10 beep Command
34. The tokens are looked up in the t comment and if found they will be replaced by the standard mathematical symbols that they correspond to before being placed in the typeset output file If not found the symbol will be output as is and a warning will be issued The tokens do not have to be active in the database although a warning will be issued if they are not declared with c or v statements The comment will stay in math mode until a second is found or the end of the comment is reached Here is an example of its use Pierce s axiom ph gt ps gt ph gt ph P P P P is not very intuitive becomes Pierce s axiom y gt Y gt y gt y is not very intuitive Note that the math symbol tokens must be surrounded by white space White space should also surround the delimiters The math mode feature also gives you a quick and easy way to generate text containing mathematical symbols independently of the intended pur pose of Metamath To do this simply create your text with grave accents surrounding your formulas after making sure that your math symbols are mapped to 4TpX symbols as described in Appendix A It is easier if you start with a database with predefined symbols such as set mm Use your grave quoted math string to replace an existing comment then typeset the statement corresponding to that comment following the instructions from the help tex command in the Metamath program You
35. a a or p statement associated with the d the two expressions that result from the substitution must have no variables in common In addition each possi ble pair of variables one from each expression must be in a d statement associated with the statement being proved This requirement forces the statement being proved to inherit the original disjoint variable restriction For example suppose u is a variable If the restriction d AB has been specified for a theorem referenced in a proof we may not substitute A with a u and B with b u because these two symbol sequences have the variable u in common Furthermore if a and b are variables we may not substitute A with a and B with b unless we have also specified d a b for the theorem being proved in other words the d property associated with a pair of variables must be effectively preserved after substitution The d statement does not mean the two variables may not be sub stituted with the same thing as you might think at first For example substituting each of A and B in the above example with identical symbol sequences consisting only of constants does not cause a disjoint variable conflict because two symbol sequences have no variables in common since they have no variables period Similarly a conflict will not occur by substi tuting the two variables in a d statement with the empty symbol sequence The d statement does not have a direct counterpart
36. a redundant axiom is omitted We extend CN with the constants var V We extend VR with an infinite set of individual metavariables x y z and denote this subset Vr We also join to CN a possibly infinite set Pr of predicates R S We associate with Pra function rnk from Pr to w and for a Pr we call rnk a the rank of the predicate a which is simply the number of arguments that the predicate has Most applications of predicate calculus will have a finite number of predicates for example set theory has the single two argument or binary predicate which is usually written with its arguments surrounding the predicate symbol rather than with the prefix notation we will use for the general case As a device to facilitate our discussion we will let Vs be any fixed one to one function from w to Vr thus Vs is any simple infinite sequence of individual metavariables with no repeating terms In this example we will not include the function symbols that are of ten part of formalizations of predicate calculus Using metalogical argu ments that are beyond the scope of our discussion it can be shown that our formalization is equivalent when functions are introduced via appropriate definitions We extend the set T defined in Example 1 with the expressions var x var y var z We extend the I above with the axiomatic statements that are the reducts of the following pre statements T wif Vx y T
37. again which happens often when building wff s The Z is placed immediately after the least significant digit letters A through T that ends the integer corresponding to the step to later be referenced The integers that the upper case letters correspond to are mapped to labels as follows If the statement being proved has m mandatory hypothe ses integers 1 through m correspond to the labels of these hypotheses in the order shown by the show statement full command i e the RPN order of the mandatory hypotheses Integers m 1 through m n corre spond to the labels enclosed in the parentheses of the compressed proof in the order that they appear where n is the number of those labels Integers m n 1 on up don t directly correspond to statement labels but point to proof steps identified with the letter Z so that these proof steps can be referenced later in the proof Integer m n 1 corresponds to the first step tagged with a Z m n 2 to the second step tagged with a Z etc When the compressed proof is converted to a normal proof the entire subproof of a step tagged with Z replaces the reference to that step For efficiency Metamath works with compressed proofs directly without converting them internally to normal proofs In addition to the usual error checking an error message is given if 1 a label in the label list in paren theses does not refer to a previous p or a statement or a non mandatory hypothesis of the stat
38. algorithm In W W Bledsoe and D W Loveland editors Automated Theorem Proving After 25 Years Proceedings of the Special Session on Automatic Theorem Proving 89th Annual Meeting of the American Mathematical Society held in Denver Colorado January 5 9 1983 pages 243 286 Providence Rhode Island 1983 American Mathemati cal Society QA76 9 A96 S64 1983 Richard Courant and Herbert Robbins Topology In James R New man editor The World of Mathematics Volume One pages 573 590 Simon and Schuster New York 1956 QA3 W67 1988 Haskell B Curry Foundations of Mathematical Logic Dover Publica tions Inc New York 1977 QA9 C976 1977 Philip J Davis and Reuben Hersh The Mathematical Experience Birkhauser Boston Boston 1981 QA8 4 D37 1982 Richard de Millo Richard Lipton and Alan Perlis Social processes and proofs of theorems and programs In Thomas Tymoczko editor New Directions in the Philosophy of Mathematics pages 267 285 Birkhauser Boston Inc Boston 1986 QA8 6 N48 1986 Robert E Edwards A Formal Background to Mathematics Springer Verlag New York 1979 QA37 2 E38 v 1a Herbert B Enderton Elements of Set Theory Academic Press Inc San Diego 1977 QA248 E5 R L Goodstein Development of Mathematical Logic Springer Verlag New York Inc New York 1971 QA9 G6554 Michael Guillen Bridges to Infinity Jeremy P Tarcher Inc Los Angeles 1983 QA93 G8 Alan G Hami
39. all of the features described in this section Currently there is no separate command to check for these errors 5 8 1 The Typesetting Comment t The HTML definitions for math symbols as well as some customization of the generated web page are specified by statements in a special typesetting comment in the input database file The typesetting comment is identified by the token t in the comment and the typesetting statements run until the next C t e HTML definitions go here In version 0 07 30 of the Metamath program there may be only one t comment in a database See the set mm database file for an extensive example of a t comment illustrating all of the features described below In the HTML definition section C style comments are recognized The main HTML specification statements are htmldef math token as HTML code 5 8 CREATING HTML OUTPUT 149 htmldef math token as HTML code htmltitle HTML code htmlhome HTML code htmlvarcolors HTML code htmlbibliography filename The htmltitle is the HTML code for a common title such as Metamath Proof Explorer The htmlhome is code for a link back to the home page The htmlvarcolors is code for a color key that appears at the bottom of each proof The file specified by filename is an HTML file that is assumed to have a lt A NAME gt tag for each bibiographic reference in the database comments For example if Monk occurs in the commen
40. and proof for example are not even mentioned in the index of the user s manual for Mathematica What is also unsatisfactory from a philosophical point of view is that there is no way to ensure the validity of the results other than by trusting the writer of each application module or tediously checking each module by hand simi lar to checking a computer program for bugs While of course extremely valuable in applied mathematics computer algebra systems tend to be of little interest to the theoretical mathematician except as aids for exploring certain specific problems Because of possible bugs trusting the output of a computer algebra sys tem for use as theorems in a proof verifier would defeat the latter s goal of rigor On the other hand a fact such that a certain relatively large number is prime while easy for a computer algebra system to derive might have a long tedious proof that could overwhelm a proof verifier One approach for linking computer algebra systems to a proof verifier while retaining the advantages of both is to add a hypothesis to each such theorem indicating its source For example a constant MAPLE could indicate the theorem came from the Maple package and would mean assuming Maple is consistent then This and many other topics concerning the formalization of math ematics are discussed in John Harrison s very interesting PhD thesis BI available on the Web 1 3 2 Automated Theorem Provers
41. as we add to the theorem an antecedent that embeds dummy and any other variables that must be distinct In a similar fashion axioms for set theory can be devised that do not require distinct variables contact me if interested Together these in principle allow all of mathematics to be developed under Metamath without a d statement although the length of the resulting theorems will grow as more and more dummy variables become required in their proofs 4 2 5 The f and e Statements Metamath has two kinds of hypotheses the f or variable type hypothesis and the e or logical hypothesis The letters f and e stand for floating roughly meaning used only if relevant and essential meaning always used respectively for reasons that will become apparent when we discuss frames in Section 4 2 7 and scoping in Section 4 2 8 The syntax of these are as follows label constant variable label e constant math symbol math symbol A hypothesis must have a label The expression in a e hypothesis consists of a constant math symbol followed by a sequence of zero or more math symbols Each math symbol including constant and variable must be a previously declared constant or variable In addition each math symbol must be active which will be covered when we discuss scoping statements in Section 4 2 8 You use a f hypothesis to specify the nature or type of a variable such as let x be an integer and use a e hyp
42. by the time the main goal is achieved the mathematician may have already figured out the proof as a by product There is also a notational problem Mathematicians are used to using very compact syntax where one or two symbols heavily de pendent on context can represent very complex concepts this is part of the hierarchy they have built up to tackle difficult problems A theorem prover on the other hand might require that a theorem be expressed in first order logic which is the logic on which most of mathematics is ultimately based but which is not ordinarily used directly because expressions can become very long Some automated theorem provers are experimental programs limited in their use to very specialized areas and the goal of many is simply research into the nature of automated theorem proving itself Finally much research remains to be done to enable them to prove very deep theorems One significant recent result was a computer proof by Larry Wos and col leagues that every Robbins algebra is a Boolean algebra New York Times Dec 10 1996 39In 1933 E V Huntington presented the following axiom system for Boolean algebra with a unary operation n and a binary operation er ty yte r e y tz 2 y z n n x y n n z n y x Herbert Robbins a student of Huntington conjectured that the last equation can be replaced with a simpler one Robbins and Huntington could not find a proof The prob
43. cate 27 61 auxiliary keyword axiom fo P EA 9 2 ET BAS BSES 125 126 Axiom of Choice Axiom of Extensionality a Axiom of Infinity 82 62 Axiom of Pairing 62 Axiom of Power Sets 62 65 Axiom of the Null Seip AR Axiom of Union 62 2 65 174 axiom scheme 35 59 63 axiom vs definition 89 SOT 107 125 126 axiomatic assertion axiomatic statement in a formal system 165 axioms for mathematics axioms in set mm axioms of equality axioms of logic axioms of predicate calculus axioms of propositional calculus 58 axioms of set theory Barrow John D basic keyword INDEX basic ee lies Behnke as Bell J L Bible biconditional binary ani 73 Blass Andrea Bledsoe W W on block 89 BI p3 on outermost 93 Boolean algebra Boolos George S bound variable 32 Bourbaki Nicolas brace notation Bunder Martin Burali Forti paradox 65 Cantor s theorem Cantor Georg 23 cardinal inaccessible xii cardinal transfinite cardinality category theory certainty Chou Shang Ching 25 class 67 70 proper class abstraction 56 a 70 174 class difference class equality class membership class variable Clifford algebras closure 165 Cohen Paul collection command keyword command line interface CLI 40 45 command qualifier comment 9
44. enough patience figure out exactly how it works even though you are not a specialist Pascal allows you do anything that a computer can do at least in principle Thus you know you have the ability in principle to follow anything that a computer program can do you just have to break it down into small enough pieces 10164 ch 1 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 5 Here s an imaginary scenario of what might happen if you naively a dopted this same view of abstract mathematics and tried to pick it up on your own in a period of time comparable to saying learning a computer programming language A Non Mathematician s Quest for Truth my daughters have been studying chemistry for several se mesters think they have learned differential and integral calculus in school and yet even today don t know why x y y is true EDMUND Lanpad Minus times minus is plus The reason for this we need not discuss W H AUDEN We ll suppose you are technically oriented professional perhaps an engi neer a computer programmer or a physicist but probably not a mathemati cian You consider yourself reasonably intelligent You did well in school learning a variety of methods and techniques in practical mathematics such as calculus and differential equations But rarely did your courses get into anything resembling modern abstract mathematics and proofs were some thing that appeared only occasionally in your textb
45. gt Y T t p AY 4 le gt 4 T E pV y e y gt a T F Exp amp 7AVa 7y The first three logical axioms statements containing introduce and effectively define logical equivalence 4 gt The last three use gt to effec tively mean is defined as C 3 7 Example 5 ZFC Set Theory Here we add to the system of Example 4 the axioms of Zermelo Fraenkel set theory with Choice For convenience we make use of the definitions in Example 4 In the CN of Example 4 which extends Example 3 we replace the symbol R with the symbol We remove from I of Example 4 the three axiomatic statements containing R and replace them with the reducts of the following D T wif x y T F t y t ez gt ye 2z 174 APPENDIX C METAMATH S FORMAL SYSTEM 9 T 2 x y gt z 27 gt 2z y Letting D a 8 DV a Vr in other words all individual vari ables must be distinct we extend I with the ZFC axioms called Exten sionality Replacement Union Power Set Regularity Infinity and Choice that are the reducts of Ext D T 9 F Va z y x E z gt y 2 Rep D T JyYz y gt z y gt Yz z x Ix x yA Vyy Un D T JaVy da y E Azx z gt yE x Pow D T 2 F daVy Va a y gt x z gt y x m Ir
46. help but be amazed at seeing 1 1 2 finally appear on page 83 of Volume II Theorem 110 643 The origin of the proof notation used by Metamath dates back to the 1950 s when the logician C A Meredith expressed his proofs in a compact notation called condensed detachment 47 This notation allows proofs to be communicated unambiguously by merely referencing the axiom rule or theorem used at each step without explicitly indicating the substitutions that have to be made to the variables in that axiom rule or theorem Ordinarily condensed detachment is more or less limited to propositional calculus The concept has been extended to first order logic in 84 making it is easy to write a small computer program to verify proofs of simple first order logic theorems A key concept behind the notation of condensed detachment is called unification which is an algorithm for determining what substitutions to variables have to be made to make two expressions match each other Uni fication was first precisely defined by the logician J A Robinson who used it in the development of a powerful theorem proving technique called the resolution principle 49 Metamath does not make use of the resolution principle which is intended for systems of first order logic Metamath s use is not restricted to first order logic and as we have mentioned it does not automatically discover proofs However unification is a key idea be hin
47. important part of a mathematician s training But is it also necessary to deny outsiders access to it Is it necessary that abstract mathematics be so hard for a layperson to grasp A computer normally is of no help whatsoever Most published proofs are actually just series of hints written in an informal style that requires con siderable knowledge of the field to understand These are the real proofs referred to by Davis and Hersh There is an implicit understanding that in principle such a proof could be converted to a complete formal proof However it is said that no one would ever attempt such a conversion even if they could because that would presumably require millions of steps Sec tion 1 1 3 Unfortunately the informal style automatically excludes the understanding of the proof by anyone who hasn t gone through the neces sary apprenticeship The best that the intelligent layperson can do is to read popular books about deep and famous results while this can be helpful it can also be misleading and the lack of detail usually leaves the reader with no ability whatsoever to explore any aspect of the field being described The statements of theorems often use sophisticated notation that makes them inaccessible to the non specialist For a non specialist who wants to achieve a deeper understanding of a proof the process of tracing definitions and lemmas back through their hierarchy quickly becomes confusing and discouraging Te
48. in an extended frame A statement of a formal system corresponds to a frame in a Metamath database C 2 THE FORMAL DESCRIPTION 165 C 2 5 Formal Systems A formal system is a triple CN VR T where T is a set of statements The members of T are called axiomatic statements Sometimes we will refer to a formal system by just IT when CN and VR are understood Given a formal system I the closurq of a pre statement D T H A is the smallest set C of expressions such that 1 TUH CC and 2 If for some axiomatic statement Diy Ty H A T and for some substitution we have a o T4 U H C C and b for all a 8 D4 for all y V o a and for all 6 V o 8 we have 7 8 D then a A C A pre statement D T H A is provable if A C i e if its assertion belongs to its closure A statement is provable if it is the reduct of a provable pre statement The universe of a formal system is the collection of all of its provable statements Note that the set of axiomatic statements T in a formal system is a subset of its universe Comment The first condition in the definition of closure simply says that the hypotheses of the pre statement are in its closure Condition 2 a says that a substitution exists that makes the mandatory hypotheses of an axiomatic statement exactly match some members of the closure This is what we explicitly demonstrate in a Metamath language proof Condit
49. in earlier versions of this specifi cation While not theoretically necessary it is imposed to make it easier to write certain parsers 4 1 SPECIFICATION OF THE METAMATH LANGUAGE 95 4 1 4 Proof Verification The sequence of labels between the and tokens in a p statement is a proof Each label in a proof must be the label of an active statement other than the p statement itself thus a label must refer either to an active hypothesis of the p statement or to an earlier assertion An expression is a sequence of math symbols A substitution map associates a set of variables with a set of expressions It is acceptable for a variable to be mapped to an expression containing it A substitution is the simultaneous replacement of all variables in one or more expressions with the expressions that the variables map to A proof is scanned in order of its label sequence If the label refers to an active hypothesis the expression in the hypothesis is pushed onto a stack If the label refers to an assertion a unique substitution must exist that when made to the mandatory hypotheses of the referenced assertion causes them to match the topmost i e most recent entries of the stack in order of occurrence of the mandatory hypotheses with the topmost stack entry matching the last mandatory hypothesis of the referenced assertion As many stack entries as there are mandatory hypotheses are then popped from the stack The same substitution is made
50. in first order logic Although the database set mm included with the Metamath soft ware package describes standard logic and set theory Metamath is actually a general purpose language for describing a wide variety of formal systems Non standard systems such as modal logic intuitionist logic higher order logic quantum logic and category theory can all be described with the Metamath language You define the symbols you prefer and tell Metamath the axioms and rules you want to start from and Metamath will verify any inferences you make from those axioms and rules A simple example of a non standard formal system is Hofstadter s MIU system whose Metamath description is presented in Appendix D Since the days of David Hilbert mathematicians have been concerned with the fact that the metalanguage used to describe mathematics may be stronger than the mathematics being described Metamath s underlying finitary constructive nature provides a good philosophical basis for studying even the weakest logics Actually the usual treatment of many non standard formal systems uses model theory or proof theory to describe these systems these theories in 30 CHAPTER 1 INTRODUCTION turn are based on standard set theory In other words a non standard formal system is defined as a set with certain properties and standard set theory is used to derive additional properties of this set The standard set theory database provided with Metamath can be
51. indented one level more than the assertion that uses it so it is easy to find the corresponding assertion by moving directly down until the indentation level decreases to one less than where you started from To the right of each is the proof step label for that proof step The statement keyword of the proof step label is listed next followed by the content of the top of the stack the most recent stack entry as it exists after that proof step is processed With a little practice you should have no trouble reading proofs displayed in this format Our simple example shows the syntax construction of a formula that might be used as part of the construction of a real proof step that de duces theorems from other theorems In standard mathematics this kind of construction is not considered a proper part of the proof at all and it certainly becomes rather boring after a while To filter out syntax con structions in the proof display the show proof command has the qualifier essential which you will probably use quite often in order to see just the real steps of the proof When verifying a proof Metamath will check that no mandatory d statement of an assertion referenced in a proof is violated when substitutions are made to the variables in the assertion For details see Section 4 1 4 or 4 2 4 4 3 1 The Concept of Unification During the course of verifying a proof when Metamath encounters an asser tion label it associates
52. is Bourbaki s language for set theory which requires 4 523 659 424 929 symbols plus 1 179 618 517 981 disambiguatory links lines connecting symbol pairs usually drawn below or above the for mula to express the number one 33 A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used Of course this is how ordinary informal mathematics is normally done anyway but with Metamath it can be done with absolute rigor and precision 1 1 4 Beauty No one shall be able to drive us from the paradise that Cantor has created for us DAVID HILBERT Mathematics possesses not only truth but some supreme beauty a beauty cold and austere like that of a sculpture BERTRAND RUssELI Euclid alone has looked on Beauty bare EDNA MiLLay23 20George S Boolos lecture at Massachusetts Institute of Technology spring 1990 21 As quoted in 40 p 131 22 53 23 As quoted in I3 p 150 16 CHAPTER 1 INTRODUCTION For most people abstract mathematics is distant strange and incom prehensible Many popular books have tried to convey some of the sense of beauty in famous theorems But even an intelligent layperson is left with only a general idea of what a theorem is about and is hardly given the tools needed to make use of it Traditionally it is only after years of arduous study that one can gr
53. it will be unique Specifically the requirement that each variable have a type specified for it with a f statement ensures the uniqueness We will illustrate this with an example Consider the following Metamath source file c gt wif vpqrs wp f wff p wq f wff q 17In the Metamath spec Section 4 1 we use the singular term substitution to refer to the set of substitutions we talk about here 116 CHAPTER 4 THE METAMATH LANGUAGE wr f wff r ws f wff s w2 a wif p gt q wnew p wff s gt r gt p ws wr wp w2 w2 This Metamath source example shows the definition and proof i e con struction of a well formed formula wff in propositional calculus You may wish to type this example into a file to experiment with the Metamath program The first two statements declare introduce the names of four constants and four variables The next four statements specify the variable types namely that each variable is assumed to be a wff Statement w2 de fines postulates a way to produce a new wff p gt q from two given wffs p and q The mandatory hypotheses of w2 are wp and wq Statement wnew claims that s gt r gt p is a wff given three wffs s r and p More precisely wnew claims that the sequence of ten symbols wff s gt r gt p is provable from previous assertions and the hypotheses of wnew Metamath does not know or care what a wff is
54. mathematical expressions as simple sequences of symbols devoid of mean ing The higher level or metamathematical notions underlying Metamath 24 Jerusalem Bible 25 Jahresberichte der Deutschen Mathematicker Vereinigung bk 2 26 As quoted in 14 p 273 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 17 are about as simple as they could possibly be Each individual step in a proof involves a single basic concept the substitution of an expression for a variable so that in principle almost anyone whether mathematician or not can completely understand how it was arrived at In one of its most basic applications Metamath can be used to develop the foundations of mathematics from the very beginning This is done in the set theory database that is provided with the Metamath package and is the subject matter of Chapter 3 Any language a metalanguage used to describe mathematics an object language must have a mathematical content of its own but it is desirable to keep this content down to a bare minimum namely that needed to make use of the inference rules speci fied by the axioms With any metalanguage there is a chicken and egg problem somewhat like circular reasoning you must assume the validity of the mathematics of the metalanguage in order to prove the validity of the mathematics of the object language The mathematical content of Meta math itself is quite limited Like the rules of a game of chess the essential concepts ar
55. mathematics is consistent free from contradictions If mathematics is consistent we will never know it but must rely on faith If mathematics is inconsistent the best we can hope for is that some clever fu ture mathematician will discover the inconsistency In this case the axioms would probably be revised slightly to eliminate the inconsistency as was done in the case of Russell s paradox but the bulk of mathematics would probably not be affected by such a discovery Russell s paradox for example did not affect most of the remarkable results achieved by 19th century and earlier mathematicians It mainly invalidated some of Gottlob Frege s work on the foundations of mathematics in the late 1800 s in fact Frege s work inspired Russell s discovery Despite the paradox Frege s work contains im portant concepts that have significantly influenced modern logic Kline s Mathematics The Loss of Certainty 28 has an interesting discussion of this topic What can be achieved with absolute certainty is the knowledge that if we assume the axioms are consistent and true then the results derived from them are true Part of the beauty of mathematics is that it is the one area of human endeavor where absolute certainty can be achieved in this sense A mathematical truth will remain such for eternity However our actual knowledge of whether a particular statement is a mathematical truth is only as certain as the correctness of the proof
56. objects called sets and from it essentially all of mathematics can be derived For example numbers can be defined as specific sets and their properties can be explored using the tools of set theory 4Number theory deals with the properties of positive and negative integers whole numbers group theory abstract algebra f analysis and topology P Even in physics Metamath could be applied to certain branches that make use of abstract mathematics such as quantum logic used to study aspects of quantum mechanics On the other hand Metamath is less suited to applications that deal primarily with intensive numeric computations Metamath does not have any built in representation of numbers instead a specific string of symbols digits must be syntactically constructed as part of any proof in which an ordinary number is used For this reason numbers in Metamath are best limited to specific constants that arise during the course of a theorem or its proof Numbers are only a tiny part of the world of abstract mathe matics The exclusion of built in numbers was a conscious decision to help achieve Metamath s simplicity and there are other software tools such as the computer algebra programs MACSYMA Mathematica and Maple specifically suited to handling numbers efficiently After learning Metamath s basic statement types any technically orient ed person mathematician or not can immediately trace any theorem proved in the langua
57. of quotes see the instructions at the end of help let in the Metamath program 5 6 9 unify Command Syntax unify step step and unify all interactive These commands available in the Proof Assistant only unify the source and target of the specified step s If you specify a specific step you will be prompted to select among the unifications that are possible If you specify all only those steps with unique unifications will be unified Optional command qualifier for unify all interactive You will be prompted to select among the unifications that are possible for any steps that do not have unique unifications Oth erwise unify all will bypass these See also set unification_timeout The default is 100000 but increas ing it to 1000000 can help difficult cases Manually assigning some or all of 5 6 CREATING PROOFS 145 the unknown variables with the let variable command also helps difficult cases 5 6 10 initialize Command Syntax initialize step step and initialize all These commands available in the Proof Assistant only de unify the target and source of a step or all steps as well as the hypotheses of the source and makes all variables in the source and the source s hypotheses unknown This command is useful to help recover from incorrect unifications that resulted from an incorrect assign let or unification choice Part or all of the command sequence delete floating_hypotheses initialize all and
58. of what these results are about but if someone asked you to prove them you wouldn t have the faintest idea of where to begin Sure you could give the same overall outline that you learned from the popular books and in a general sort of way you do have an understanding But deep down inside you know that there is a rigor that is missing that probably there are many subtle steps and pitfalls along the way and ultimately it seems you have to place your trust in the experts in the field You don t like this you want to be able to verify these results for yourself So where do you go next As a first step you decide to look up some of the original papers on the theorems you are curious about or better obtain some standard textbooks in the field You look up a theorem you want to understand Sure enough it s there but it s expressed with strange terms and odd symbols that mean absolutely nothing to you It might as well be written in a foreign language you ve never seen before whose symbols are totally alien You look at the proof and you haven t the foggiest notion what each step means much less how one step follows from another Well obviously you have a lot to learn if you want to understand this stuff You feel that you could probably understand it by going back to college for another three to six years and getting a math degree But that does not fit in with your career and the other things in your life and would serve no pr
59. orem 7 13 of Quine p 51 prex p F A B eV Next we will list some famous or important theorems that are proved in the set mm database None of them except omex require the Axiom of 3 6 A THEOREM SAMPLER 79 Infinity as you can verify with the show trace_back Metamath command The resolution of Russell s paradox There exists no set containing the set of all sets which are not members of themselves Proposition 4 14 of Takeuti and Zaring p 14 ru p F r gt daz y ryey Cantor s theorem No set can be mapped onto its power set Compare Theorem 6B b of Enderton p 132 canth 1 e FAEV canth p 7AF A PA onto The Burali Forti paradox No set contains all ordinal numbers Enderton p 194 Burali Forti was one person not two onpre p f7 OneV Peano s postulates for arithmetic Proposition 7 30 of Takeuti and Zaring pp 42 43 The objects being described are the members of w i e the natural numbers 0 1 2 The successor operation suc means plus one peanot says that 0 which is defined as the empty set is a natural number peano2 says that if A is a natural number so is A 1 peano3 says that 0 is not the successor of any natural number peano4 says that two natural numbers are equal if and only if their successors are equal peanod is essentially the same as mathematical induction peanot p F Ew peano2 p AeG w sucAew peano3 p A w gt nrsucA 2 F peano4 p H AcwAB
60. physically adjacent in a database we will cover this in our discussion of scoping statements in Section 4 2 8 A frame has the following properties 1 The set of variables contained in its f statements must be identical to the set of variables contained its e a and or p statements In other words each variable in a e a or p statement must have an associated variable type defined for it in a f statement 2 No two f statements may contain the same variable 3 Each of the two variables in any d statement must occur in a f statement 4 Any f statement must occur before a e or d statement in which its variable occurs The first property determines the set of variables occurring in a frame These are the mandatory variables of the frame The second property tells us there must be only one type specified for a variable The third prop erty determines which d statements belong to the frame The last property is not a theoretical requirement but it makes parsing of the database easier For our examples we assume our database has the following declarations v PQR c gt l wff The following sequence of statements describing the modus ponens in ference rule is an example of a frame wp f wff P wq f wff Q maj e l P gt Q min e P mp a l Q The following sequence of statements is not a frame because R does not occur in the e s or the a wp f wff P
61. preserved after addition to both sides axltadd p F AERABERACER gt A lt B gt C A lt C B 26 The product of two positive reals is positive axmulgtO p F AERABER gt 0 lt A 0 lt B gt 0 lt A B 27 A complex number can be expressed in terms of two reals axcnre p H AcC gt J3reRJyERA x y i 28 A non empty bounded above set of reals has a supremum axsup p F ACRAA AJzrERYyEAy lt zr gt JzrER YyEA na lt yAVyER y lt a7dzeAy lt z This completes the set of axioms for real and complex numbers You may wish to look at how subtraction division and decimal numbers are defined in set mm and for fun look at the proof of 2 2 4 theorem 2p2e4 in set mm In set mm we define the non negative integers N the integers Z and the rationals Q as subsets of R This leads the nice inclusion NC ZC QC R C C giving us a uniform framework in which for example a property such as commutativity of complex number addition automatically applies to integers The natural numbers N are different from the set w we defined earlier but both satisfy Peano s postulates Complex Number Axioms in Analysis Texts Most analysis texts construct complex numbers as ordered pairs of reals leading to construction dependent properties that satisfy these axioms but are not stated in their pure form This is also done in set mm but our axioms are extracted from that construction Other texts will
62. propositional calculus the Principle of Identity which is proved directly from the axioms We ll look at the statement then its proof MM gt show statement idi full Statement 36 is located on line 968 of the file set mm Principle of identity Theorem 2 08 of Whitehead and Russell This version is proved directly from the axioms for demonstration purposes id1 p ph gt ph Its mandatory hypotheses in RPN order are wph f wff ph Its optional hypotheses are wps wch wth wet The statement and its hypotheses require the variables ph These additional variables are allowed in its proof ps ch th et The variables it contains are ph The optional variables ps ch etc are available for use in a proof of this statement if we wish and were we to do so we would make use of optional hypotheses wps wch etc See Section for the meaning of optional hypothesis The reason these show up in the statement display is that statement id1 happens to be in their scope see Section for the definition of scope but in fact in propositional calculus we will never make use of optional hypotheses or variables This becomes important after quantifiers are introduced where dummy variables are often needed in the middle of a proof 86 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED Let s look at the proof of statement id1 We ll suppress the non essential steps that construct the wffs MM gt show pro
63. result of the calculation For example the input sequence 2 2 on an RPN calculator results in 4 and the input sequence 1 2 2 results in 1 4 which results in 5 Understanding how RPN is processed involves the concept of a stack which can be thought of as a set of temporary memory locations that hold in termediate results When Metamath encounters a hypothesis label it places or pushes the math symbol sequence of the hypothesis onto the stack When Metamath encounters an assertion label it associates the most recent stack entries with the mandatory hypotheses of the assertion in the order where the most recent stack entry is associated with the last mandatory hypothesis of the assertion It then determines what substitutions have to be made into the variables of the assertion s mandatory hypotheses to make them identi cal to the associated stack entries It then makes those same substitutions into the assertion itself Finally Metamath removes or pops the matched hypotheses from the stack and pushes the substituted assertion onto the stack For the purpose of matching the mandatory hypothesis to the most recent stack entries whether a hypothesis is a e or f statement is irrelevant The only important thing is that a set of substitutiond exist that allow a match and if they don t the proof verifier will let you know with an error message The Metamath language is specified in such a way that if a set of substitutions exists
64. so that you can examine any errors later on To open a log file type If a directory path is needed on Unix you should enclose the path file name in quotes to prevent Metamath from thinking that the in the path name is a command qualifier e g read db set mm Quotes are optional when there is no ambiguity 42 CHAPTER 2 USING THE METAMATH PROGRAM MM gt open log abc log This will open a file called abc log and everything that appears on the screen from this point on will be stored in this file The name of the log file is arbitrary To close the log file type MM gt close log Several commands let you examine what s inside of your database Sec tion 3 8 has an overview of some useful ones The show labels command lets you see what statement labels exist A matches any combination of characters and t refers to all labels starting with the letter t The al1 is a command qualifier that tells Metamath to include labels of hypotheses To see the syntax explained type help show labels Type MM gt show labels t all Metamath will respond with The statement number label and type are shown 3 tt f 4 tr f 5 ts f 8 tze a 9 tpl a 19 thi p You can use the show statement command to get information about a particular statement For example you can get information about the statement with label mp by typing MM gt show statement mp full Metamath will respond with Statement 17 is located on line 43 o
65. system is logically equivalent to the more typical textbook systems that do have these primi tive notions if we introduce these notions with appropriate definitions and metalogic We could also define axioms for such systems directly although the recursive definitions of free variable and proper substitution would be messy and awkward to work with Instead we mention two devices that can be used in practice to mimic these notions 1 Instead of introducing special notation to express as a logical hypothesis where x is not free in yp we can use the logical hypothesis H p gt Vx pP 2 It can be shown that the wf x y gt y A Jz x y with the usual definitions of and J see Example 4 below is logically equivalent to the wff that results from proper substitution of y for x in y This works whether or not x and y are distinct C 3 4 Metalogical Completeness In the system of Example 2 the following are provable pre statements and their reducts are provable statements x y T F WIE y 2 7 2 Fk Nrg z whereas the following pre statement is not to my knowledge provable but in any case we will pretend it s not for sake of illustration 9 T F Wasa y 5This is a slightly weaker requirement than where is not free in vy If we let y be x a we have the theorem x x Yx x x which satisfies the hypothesis even though z is free in x x In a case like
66. the mandatory hypotheses of the assertion with the top entries of the RPN stack Metamath then determines what substitu tions it must make to the variables in the assertion s mandatory hypotheses in order for these hypotheses to become identical to their corresponding stack entries This process is called unification We also informally use the term unification to refer to a set of substitutions that results from the process as in two unifications are possible After the substitutions are made the hypotheses are said the be unified If no such substitutions are possible Metamath will consider the proof incorrect and notify you with an error message The general algorithm for unification described in the literature is some what complex However in the case of Metamath it is trivial because of the requirement that each variable have its type specified with a f hypoth esis and that each f hypothesis have the restricted syntax of a constant followed by a variable The constant in the f hypothesis must match the first symbol of the corresponding RPN stack entry which will be also be a 4 4 EXTENSIONS TO THE METAMATH LANGUAGE 119 constant so only possible match for the variable in the f hypothesis is the sequence of symbols in the stack entry after the initial constant In the Proof Assistant a more general unification algorithm is used While a proof is being developed sometimes not enough information is avail able to d
67. these steps out by itself and we will not have to worry about them Therefore from here on we will display only the essential hypotheses i e those steps that correspond to traditional formal proofs MM PA gt show new_proof lemmon essential 37 2 4 2 gt t t 5 3 4 mp a l t t Unknown steps 3 and 4 are the ones we must focus on They correspond to the minor and major premises of the modus ponens rule We will assign them as follows Notice that because of the step renumbering that takes place after an assignment it is advantageous to assign unknown steps in reverse order because earlier steps will not get renumbered MM PA gt assign 4 mp To undo the assignment DELETE STEP 8 and INITIALIZE UNIFY if needed 2 4 YOUR FIRST PROOF 49 3 min 2 6 min 4 7 maj 4 gt 2 gt t t We are now going to describe an obscure feature that you will probably never use but should be aware of The Metamath language allows empty symbol sequences to be substituted for variables but in most formal systems this feature is never used One of the few examples where is it used is the The MIU system described in Appendix D But such systems are rare and by default this feature is turned off in the Proof Assistant It is always allowed for verify proof Let us turn it on and see what happens MM PA gt set empty_substitution on Substitutions with empty symbol sequences is now allowed Wit
68. they do not differ either from the axioms or from theo rems i e from the theses added to the system on the basis of the rule of substitution or the rule of detachment modus ponens Once definitions have been accepted as theses of the system it becomes necessary to consider them as true propositions in the same sense in which axioms are true 32 Let us look at some simple examples of definitions in propositional cal culus Consider the definition of logical OR disjunction P V Q denotes P Q not P implies Q It is very easy to recognize a statement mak ing use of this definition because it introduces the new symbol V that did not previously exist in the language It is easy to see that no new theorems of the original language will result from this definition Next consider a definition that eliminates parentheses P gt Q gt R denotes P Q gt R This is more subtle because no new symbols are introduced The reason this definition is considered proper is that no new symbol sequences that are valid wffs well formed formulas in the original language will result from the definition since P Q gt R is not a wff in the original language Here we implicitly make use of the fact that there is a decision procedure that allows us to determine whether or not a symbol sequence is a wff and this fact allows us to use symbol sequences that are not wffs to represent other things such as wffs by means of the d
69. this we say that x is effectively not free in x z since x 7v is logically equivalent to Vz x x in which z is bound C 3 EXAMPLES OF FORMAL SYSTEMS 171 In other words we can prove Va 7x y where x and y are distinct and separately prove Va 7xa x but we can t prove the combined general case Vax y that has no proviso Now this does not compromise logi cal completeness because the variables are really metavariables and the two provable cases together cover all possible cases The third case can be con sidered a metatheorem whose direct proof using the system of Example 2 lies outside the capability of the formal system Also in the system of Example 2 the following pre statement is not to my knowledge provable again a conjecture that we will pretend to be the case 9 T F Way gt Instead we can only prove specific cases of y involving individual metavari ables and by induction on formula length prove as a metatheorem outside of our formal system the general statement above The details of this proof are found in 25 There does however exist a system of predicate calculus in which all such simple metatheorems as those above can be proved directly and we present it in Example 3 A simple metatheorem is any statement of the formal system of Example 2 where all distinct variable restrictions consist of either two individual metavariables or an individual metavariable and a wf
70. to the referenced assertion and the result is pushed onto the stack After the last label in the proof is processed the stack must have a single entry that matches the expression in the p statement containing the proof A proof may contain a in place of a label to indicate an unknown step Section 4 4 5 A proof verifier may ignore any proof containing but should warn the user that the proof is incomplete A compressed proof is an alternate proof notation described in Appen dix B also see references to compressed proof in the Index Compressed proofs are a Metamath language extension which a complete proof verifier should be able to parse and verify Verifying Disjoint Variable Restrictions Each substitution made in a proof must be checked to verify that any disjoint variable restrictions are satisfied as follows If two variables replaced by a substitution exist in a mandatory d state ment of the assertion referenced the two expressions resulting from the substitution must meet satisfy the following conditions First the two ex pressions must have no variables in common Second each possible pair of variables one from each expression must exist in an active d statement of the p statement containing the proof This ends the specification of the Metamath language 96 CHAPTER 4 THE METAMATH LANGUAGE 4 2 The Basic Keywords Our expository material begins here Like most computer languages Metamath takes its i
71. try to explain the details or motivation for this we refer you to the textbooks that are mentioned in the descriptions The set mm file has bibliographic references for the text references Their proofs often embody important concepts you may wish to explore with the Metamath program see Section 3 8 All the symbols that are used here are defined in Section 3 4 For brevity we haven t included the d restrictions or f hypothesis for these theorems when you are uncertain consult the set mm database Our first theorem is not very deep but provides us with a notational device that is frequently used It allows us to use the expression A V as a compact way of saying that class A exists i e is a set Two ways to say A is a set A is a member of the universe V if and only if A exists i e there exists a set equal to A Theorem 6 9 of Quine p 43 isset p AGCVH dr r A Next we prove the axioms of standard ZF set theory that were missing from our axiom system From our point of view they are theorems since they can be derived from the other axioms Axiom of Separation Aussonderung proved from the other axioms of ZF set theory Compare Exercise 4 of Takeuti and Zaring p 22 inex1 1 e FAEV inex p F ANB EV Axiom of the Null Set proved from the other axioms of ZF set theory Corol lary 5 16 of Takeuti and Zaring p 20 Oex p Fk GEV The Axiom of Pairing proved from the other axioms of ZF set theory The
72. unify all interactive will cover from incorrect unifications See also unify and delete 5 6 11 delete Command Syntax delete step step and delete all Warning dangerous and delete floating_hypotheses These commands are available in the Proof Assistant only The delete step command deletes the proof tree section that branches off of the speci fied step and makes the step become unknown delete all is equivalent to delete step step where step is the last step in the proof i e the beginning of the proof tree There is currently no undo command and you cannot reverse a delete The best you can do is salvage your last save new_proof by exiting and reentering the Proof Assistant For this reason it is important to keep a log file open to record your work and to do save new_proof frequently especially before delete delete floating_hypotheses will delete all sections of the proof that branch off of f statements It is sometimes useful to do this before an initialize command to recover from an error Note that once a proof step with a f hypothesis as the target is completely known the improve command can usually fill in the proof for that step Unlike the deletion of logical steps delete floating_hypotheses is a relatively safe command that is usually easy to recover from 5 6 12 improve Command Syntax improve step depth number no_distinct and improve first depth number no_distinct and improve last depth nu
73. wif x y T F y E Ve y T F Walp gt Y gt Vz p gt Ve z p T F p gt Yz y 2 y T Ware y T 9 x z gt y gt z y T 9 y z gt z y gt z 2 These are the axioms not involving predicate symbols The first two state ments extend the definition of a wff The third is the rule of generalization The fifth states in effect For a wff y and variable x p gt Vay pro vided that x does not occur in vy The sixth states For a variables x and C 3 EXAMPLES OF FORMAL SYSTEMS 169 y F Va7a y provided that x and y are distinct This proviso is not necessary but was included by Tarski to weaken the axiom and still show that the system is logically complete Finally for each predicate symbol a Pr we add to the an axiomatic statement extending the definition of wff that is the reduct of the following pre statement a T wif a gt Vs ft rnk a and for each a Pr and each n lt rnk a we add to T an equality axiom that is the reduct of the following pre statement 2 T F Vsn Vstnk a gt a Vs rnk a gt a o Vsf na Vsink a c Vs rnk a n 1 gt where denotes function domain restriction and denotes set difference Recall that a subscript on Vs denotes one of its terms In the above two axiom sets commas are placed between successi
74. will then probably want to edit the resulting file with a text editor to fine tune it to your exact needs 4 4 EXTENSIONS TO THE METAMATH LANGUAGE 121 Label References in Comments Outside of math mode a tilde indicates to Metamath s output processor that the token that follows i e the characters up to the next white space represents a statement label Whether or not the token is an actual state ment label is not checked and the token does not have to have the correct syntax for a label no error messages will be produced The only effect of the label mode on the output is that typewriter font will be used for the tokens that are placed in the ATEX output file On the other hand the tokens after the tilde must be actual labels for correct output of HTML described in the next section and error messages will be issued during that output if they aren t 4 4 2 Comment Markup Notation for HTML The automated generation of HTML web pages Section 5 8 is new in version 0 07 30 of the Metamath program The comment markup notation for ATEX also applies to HTML and some additional features specifically for HTML were added These markup features including those reused from TFX are the following lt math symbol math symbol math symbols enclosed in grave ac cents i e backticks Use graphical math symbols in the HTML output Inside of a math symbol means literal label tilde followed by a label Us
75. with a set of class variables A B C We extend the T of Example 5 with class A class B class C To introduce our definitions we add to I of Example 5 the axiomatic statements that are the reducts of the following pre statements T class x T class x y T 2 wif A B T 2 wif A B C 4 METAMATH AS A FORMAL SYSTEM 175 Ab T 2 F y zlp x y gt p AAt z yA Eq 2 A a BH T HF A B amp Vr x A amp z B El x A a B T F A BS 3z x A z E B Here we say that an individual variable is a class z y is a class and we extend the definition of a wff to include class equality and membership Axiom Ab defines membership of a variable in a class abstraction the right hand side can be read as the wff that results from proper substitution of y for x in oP Axioms Eq and El extend the meaning of the existing equality and membership connectives This is potentially dangerous and requires careful justification For example from Eq we can derive the Axiom of Extensionality with predicate logic alone thus in principle we should include the Axiom of Extensionality as a logical hypothesis However we do not bother to do this since we have already presupposed that axiom earlier The distinct variable restrictions should be read where x does not occur in A or B We typically do this when the right hand si
76. wq f wff Q wr f wff R maj e P gt Q min e P mp a l Q The following sequence of statements is not a frame because Q does not occur in a f 110 CHAPTER 4 THE METAMATH LANGUAGE wp f wff P maj e l P gt Q min e P mp a l Q The following sequence of statements is not a frame because the a state ment is not the last one wp f wff P wq f wff Q maj e P gt Q mp a l Q min e P Associated with a frame is a sequence of mandatory hypotheses This is simply the set of all f and e statements in the frame in the order they appear A frame can be referenced in a later proof using the label of the a or p assertion statement and the proof makes an assignment to each mandatory hypothesis in the order in which it appears This means the order of the hypotheses once chosen must not be changed so as not to affect later proofs referencing the frame s assertion statement The Metamath proof verifier will of course flag an error if a proof becomes incorrect by doing this Since proofs make use of Reverse Polish notation described in Section 4 3 we call this order the RPN order of the hypotheses Note that d statements are not part of the set of mandatory hypotheses and their order doesn t matter as long as they satisfy the fourth property for a frame described above The d statements specify restrictions on variables that must be sati
77. you in effect instruct Metamath what symbol substitutions to make in previous axioms or theorems and join a sequence of them together to result in the desired theorem This kind of symbol manipulation captures the essence of mathematics at a preaxiomatic level Metamath and Mathematical Literature In advanced mathematical literature proofs are usually presented in the form of short outlines that often only an expert can follow This is partly out of a desire for brevity but it would also be unwise even if it were practical to present proofs in complete formal detail since the overall picture would be lost A solution I envision that would allow mathematics to remain acceptable to the expert yet increase its accessibility to non specialists consists of a combination of the traditional short informal proof in print accompanied by a complete formal proof stored in a computer database In an analogy with a computer program the informal proof is like a set of comments that describe the overall reasoning and content of the proof whereas the com puter database is like the actual program and provides a means for anyone even a non expert to follow the proof in as much detail as desired exploring it back through layers of theorems like subroutines that call other subrou tines all the way back to the axioms of the theory In addition the computer database would have the advantage of providing absolute assurance that the proof is correct sin
78. 0 31 al al t 0 t gt t 0 t gt t t 33 15 24 26 32 mp a l t 0 t gt t t 34 5 8 10 33 mp a l t t The lemmon command qualifier specifies what is known as a Lemmon style display Omitting the lemmon qualifier results in a tree style proof see p for an example that is somewhat less explicit but easier to follow once you get used to it The first number on each line is the step number of the proof Any numbers that follow are step numbers assigned to the hypotheses of the 44 CHAPTER 2 USING THE METAMATH PROGRAM statement in referenced by that step Next is the label of the statement referenced by the step The statement type of the statement referenced comes next followed by the math symbol string constructed by the proof up to that step The last step 34 contains the statement that is being proved Looking at a small piece of the proof notice that steps 3 and 4 have established that t O and t are terms and step 5 makes use of steps 3 and 4 to establish that t 0 t is a wff Let us let Metamath itself tell us in detail what is happening in step 5 Note that the target hypothesis refers to where step 5 is eventually used i e in step 34 MM gt show proof thi detailed_step 5 Proof step 5 wp weq a wff t 0 t This step assigns source weq a to target wp f The source assertion requires the hypotheses tt f step 3 and tr f step 4 The parent assertion of the target hypot
79. 2 4 and syntax rules that describe how symbols may be combined to form a legal expression called a well formed formula or wff pronounced whiff some starting wffs called axioms and inference rules that describe how theorems may be derived proved from the axioms A theorem is a mathematical fact such as 2 2 4 Strictly speaking even an obvious fact such as this must be proved from axioms to be formally acceptable to a mathematician ix x PREFACE viewpoint even if you have no formal abstract mathematics background By using a single consistent notation to express proofs once you grasp its basic concepts Metamath provides you with the ability to immediately follow and dissect proofs even in totally unfamiliar areas Of course just being able follow a proof will not necessarily give you an intuitive familiarity with mathematics Memorizing the rules of chess does not give you the ability to appreciate the game of a master and knowing how the notes on a musical score map to piano keys does not give you the ability to hear in your head how it would sound But each of these can be a first step Metamath allows you to explore proofs in the sense that you can see the theorem referenced at any step expanded in as much detail as you want right down to the underlying axioms of logic and set theory in the case of the set theory database provided While Metamath will not replace the higher level understanding that can only be
80. 3 T9 comments markup notation compact proof 189 composition compound declaration compressed proof 124 computer algebra system 3 211 computer program bugs vilil 8 wfe computers and pure mathematics concatenation condensed detachment and first order logic conjunction A connective consistent theory constant x 38 40 63 93 99 13 in a formal system in predicate calculus 64 constant declaration 38 Bar 99 constant prefixed expression 163 constant variable pair constructive language constructivism continuum hypothesis Courant Richard 23 Courier font Coxeter H S M 24 cranks creative definition cross product Curry Haskell B database Davis Phillip J de Millo Richard de Saint Exupery Antoine decidable theory decision one declaration deduction theorem definiendum definiens definition 15 172 125 126 creative 190 INDEX eliminability finite n termed sequence 162 proper finite induction 79 disjoint sets first order logic 11 20 26 31 56 disjoint variables completeness disjoint variable restriction floating hypothesis 39 formal logic formal DG in a formal system disjunction V 69 distinct variables 100 103 as domain formal system dummy variable 85 165 17 formalism foundations of mathematics founded re
81. 717 697 707 November 1980 QA A5125 George G Szpiro Poincar s Prize The Hundred Year Quest to Solve One of Math s Greatest Puzzles Penguin Books Ltd London 2007 QA43 8985 2007 Gaisi Takeuti and Wilson M Zaring Introduction to Axiomatic Set Theory Springer Verlag New York Inc New York second edition 1982 QA248 T136 1982 Alfred Tarski What is elementary geometry In Leon Henkin Patrick Suppes and Alfred Tarski editors The Axiomatic Method with Special Reference to Geometry and Physics Proceedings of an International Symposium held at the University of California Berkeley December 26 1957 January 4 1958 pages 16 29 Amsterdam 1959 North Holland Publishing Company Alfred Tarski A simplified formalization of predicate logic with iden tity Archiv f r Mathematische Logik und Grundlagenforschung 7 61 79 1965 QA A673 Thomas Tymoczko editor New Directions in the Philosophy of Math ematics Birkh user Boston Inc Boston 1986 QA8 6 N48 1986 Hao Wang Theory and practice in mathematics In Thomas Tymoczko editor New Directions in the Philosophy of Mathematics pages 129 152 Birkhauser Boston Inc Boston 1986 QA8 6 N48 1986 BIBLIOGRAPHY Alfred North Whitehead An Introduction to Mathematics 1911 Alfred North Whitehead and Bertrand Russell Principia Mathematica Cambridge University Press Cambridge second edition 1927 3 vols QA9 W592 1927 Ste
82. A B ER 10 Addition of complex numbers is commutative axaddcom p H AECABEC gt A B B A4A 11 Multiplication of complex numbers is commutative axmulcom p AECCABEC gt A B B A 12 Addition of complex numbers is associative axaddass p H AECABECACEC gt A B C A B C 13 Multiplication of complex numbers is associative axmulass p AECCABECACEC gt A B C A B C 14 Multiplication distributes over addition for complex numbers axdistr p AECCABECACEC gt A B4C A B 4 A C 15 One and zero are distinct axineO p F140 16 Zero is an identity element for addition ax0id p f AEC gt A 0 A 17 One is an identity element for multiplication axlid p AE C gt A 1 A 18 Every complex number has a negative 82 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED axnegex p AGCC gt 4xeEC A z2 0 19 Every nonzero complex number has a reciprocal axrecex p AGCC gt A40 gt 42 C A 2 1 20 Every real number has a negative axrnegex p ACR gt JAreER A4 2z 0 21 Every nonzero real number has a reciprocal axrrecex p AGCR gt A40542ER A 2 1 22 The square of i equals 1 expressed as i squared plus 1 is 0 axi2m1 p i 2 1 0 23 Ordering on reals satisfies strict trichotomy axlttri p F AERABER gt A lt B amp A BVB lt A 24 Ordering on reals is transitive axlttrn p F AERABERACER gt A lt BAB lt C gt A lt C 25 Ordering on reals is
83. A mathematical theory is decidable if a mechanical method or algorithm exists that is guaranteed to determine whether or not a particular formula is a theorem Among the few theories that are decidable is elementary ge ometry as was shown by a classic result of logician Alfred Tarski in 1948 But most theories including elementary arithmetic are undecidable This fact contributes to keeping mathematics alive and well since mathe 37Two examples illustrate why the knowledge database of computer algebra systems should sometimes be regarded with a certain caution If you ask Mathematica version 3 0 to Solve x n y n zn n it will respond with n gt 2 n gt 1 n gt 1 n gt 2 In other words Mathematica seems to know that Fermat s Last Theorem is true At the time this version of Mathematica was released this fact was unknown If you ask Maple to solve x72 27x then simplify it returns the solution set 2 4 apparently unaware that 0 7666647 is also a solution 38Tarski s result actually applies to a subset of the geometry discussed in elementary textbooks This subset includes most of what would be considered elementary geometry but it is not powerful enough to express among other things the notions of the circum ference and area of a circle Extending the theory in a way that includes notions such as these makes the theory undecidable as was also shown by Tarski Tarski s algorithm is far t
84. ETAMATH LANGUAGE for definitions In set mm we adopt the convention of prefixing axiom labels with ax and definition labels with df The results that can be derived with the Metamath language are only as good as the a statements used as their starting point We cannot stress this too strongly For example Metamath will not prevent you from spec ifying z x as an axiom of logic It is essential that you scrutinize all a statements with great care Because they are a source of potential pitfalls it is best not to add new ones usually new definitions casually rather you should carefully evaluate each one s necessity and advantages Once you have in place all of the basic axioms and rules of a mathe matical theory the only a statements that you will be adding will be what are ordinarily called definitions In principle definitions should be in some sense eliminable from the language of a theory according to some conven tion usually involving logical equivalence or equality The most common convention is that any formula that was syntactically valid but not provable before the definition was introduced will not become provable after the def inition is introduced In an ideal world definitions should not be present at all if one is to have absolute confidence in a mathematical result How ever they are necessary to make mathematics practical for otherwise the resulting formulas would be extremely long and incomprehensible Since th
85. Ew gt suc A su B A B F EAANYrEw rEA gt suczrEeA gt wCA peano5 p Finite Induction mathematical induction The first hypothesis is the basis and the second is the induction hypothesis Theorem Schema 22 of Suppes p 136 findes 1 e H xz p findes 2 e H rEw gt y gt sucz x e findes p H rEw gt g Transfinite Induction with explicit substitution The first hypothesis is the basis the second is the induction hypothesis for successors and the third is the induction hypothesis for limit ordinals Theorem Schema 4 of Suppes p 197 tfindes 1 e F xz p tfindes 2 e x On gt y suca 2 tfindes 3 e Limy gt Vreyypol y 2zly 80 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED tfindes p re On gt y Principle of Transfinite Recursion Theorem 7 41 of Takeuti and Zaring p 47 Transfinite recursion is the key theorem that allows arithmetic of ordinals to be rigorously defined and has many other important uses as well Hypotheses tfr 1 and tfr 2 specify a certain proper class F The complicated definition of F is not important in itself what is important is that there be such an F with the required properties and we show this by displaying F explicitly tfr1 states that F is a function whose domain is the set of ordinal numbers tfr2 states that any value of F is completely determined by its previous values and the values of an auxiliary function G tfr3 states that F is unique
86. FACE XV find Feferman s arguments persuasive Mac Lane may be right but when I think about category theory I do it la Grothendieck By the way Mizar adds the axiom there is a proper class of inaccessibles precisely so as to do category theory The most current information on the Metamath program and databases can always be found at http metamath org Note Added June 24 2006 The Metamath spec was restricted slightly to make parsers easier to write See the footnote on p Note Added March 10 2007 I am grateful to Anthony Williams for writing the ATX package called realref sty and contributing it to the public domain This package allows the internal hyperlinks in a PDF file to anchor to specific page numbers instead of just section titles making the navigation of the PDF file for this book much more pleasant and logical A typographical error found by Martin Kiselkov was corrected A con fusing remark about unification was deleted per suggestion of Mel O Cat Note Added May 27 2009 Several typos found by Kim Sparre were corrected A note was added that the Poincar conjecture has been proved p 24h Note Added Nov 17 2014 The statement of the Schr der Bernstein theorem was corrected in Sec tion Thanks to Bob Solovay for pointing out the error Xvi PREFACE Chapter 1 Introduction LM No no There s nothing subjective about it Everybody knows what a proof is Just read some books tak
87. FINITIONS 125 4 5 Appendix Axioms vs Definitions Metamath makes no distinction between axioms and definitions The a statement is used for both At first this may seem puzzling In the minds of many mathematicians the distinction is clear even obvious and hardly worth discussing A definition is considered to be merely an abbreviation that can be replaced by the expression for which it stands although unless one actually does this to be precise that one should say that a theorem is a consequence of the axioms and the definitions that are used in the formulation of the theorem 4 p 20 What is a definition In its simplest form a definition introduces a new symbol and provides an unambiguous rule to transform an expression containing the new symbol to one without it The concept of a proper definition as opposed to a creative definition that is usually agreed upon is 1 the definition should not strengthen the language and 2 any symbols introduced by the definition should be eliminable from the language 42 In other words they are mere typographical conveniences that do not belong to the system and are theoretically superfluous This may seem obvious but in fact the nature of definitions can be subtle sometimes requiring difficult metatheorems to establish that they are not creative A more conservative stance was taken by logician S Le niewski Le niewski regards definitions as theses of the system In this respect
88. METAMATH S FORMAL SYSTEM We also note that once established as provable a statement may be considered to acquire the same status as an axiomatic statement because if the set of axiomatic statements is extended with a provable statement the universe of the formal system remains unchanged provided that VR is infinite In practice this means we can build a hierarchy of provable statements to more efficiently establish additional provable statements This is what we do in Metamath when we allow proofs to reference previous p statements as well as previous a statements C 3 Examples of Formal Systems Relationship to Metamath The examples in this section except Exam ple 2 are for the most part exact equivalents of the development in the set theory database set mm You may want to compare Examples 1 3 and 5 to Section E Example 4 to Sections and 3 4 2 and Example 6 to 3 4 3 Section C 3 1 Example 1 Propositional Calculus Classical propositional calculus can be described by the following formal system We assume the set of variables is infinite Rather than denoting the constants and variables by co c1 and vo v1 for readability we will instead use more conventional symbols with the understanding of course that they denote distinct primitive objects Also for readability we may omit commas between successive terms of a sequence thus wf y denotes wif p Let CN wf BS VR y x T wff p
89. Metamath A Computer Language for Pure Mathematics Norman Megill PUBLIC DOMAIN This book has been released into the Public Domain by Norman Megill on March 10 2007 per the Creative Commons Public Domain Dedication fnstp creativecomons org Ticenses publicdonain The public domain release applies worldwide In case this is not legally possible the right is granted to use the work for any purpose without any conditions unless such conditions are required by law Several short attributed quotations from copyrighted works appear in this book under the fair use provision of Section 107 of the United States Copyright Act Title 17 of the United States Code The public domain status of this book is not applicable to those quotations Any trademarks used in this book are the property of their owners ISBN 978 1 4116 3724 5 Lulu Press Morrisville North Carolina USA Norman Megill 19 Locke Lane Lexington MA 02420 E mail address nm alum mit edu http metamath org Contents pan iu E Ging eRe ee Sth a Be oS ee eS 1 1 1 Is Mathematics User Friendly 1 1 2 Mathematics and the Non Specialist 1 1 3 An Impossible Dream LIA Beatty o e e goats a a s Se ea eS ao Ga ae a a LIS Simplicity a re d ae kod ee a ead E a a 1 2 Computers and Mathematicians 1 2 1 Trusting the Computer oaa 1 2 2 Trusting the Mathematician ooa 1 3 1 Computer Algebra System
90. OGRAM MM PA gt show new_proof not_unified 4 min a l 6 a2 9 0 9 Later we can unify this with the unify all interactive command The important point to remember is that occasionally you will be pre sented with several unification choices while entering a proof when the program determines that there is not enough information yet to make an unambiguous choice automatically and this can happen even with set empty_substitution turned off Usually it is obvious by inspection which choice is correct since incorrect ones will tend to be meaningless fragments of wffs In addition the correct choice will usually be the first one presented unlike our example above Enough of this digression Let us go back the the default setting MM PA gt set empty_substitution off The ability to substitute empty expressions for variables has been turned off Note that this may make the Proof Assistant too restrictive in some cases If we delete the proof start over and get to the point where we digressed above there will no longer be an ambiguous unification MM PA gt assign 3 a2 To undo the assignment DELETE STEP 4 and INITIALIZE UNIFY if needed 7 min 4 8 maj 4 gt 5 0 5 gt t t Let us look at our proof so far and continue MM PA gt show new_proof lemmon 4 a2 a Il 5 0 5 7 4 8 7 lI 4 gt 5 0 5 gt t t 9 7 8 mp a l 5 0 5 gt t t 10 4 9
91. P rule of inference It allows us to obtain new theorems from old ones The proof of a theorem is a sequence of one or more theorems each of which is either an axiom or the result of modus ponens applied to two previous theorems in the sequence and the last of which is the theorem being proved The theorem we will prove for our example is very simple t t The proof of our theorem follows Study it carefully until you feel sure you un derstand it 1 0 t by axiom A2 2 t 0 t by axiom A2 3 0 t t 0 t gt t t by axiom Al 4 t 0 t gt t t by MP applied to steps 2 and 3 bs tet by MP applied to steps 1 and 4 You may wonder why step 1 is repeated twice This is not necessary in the formal language we have defined but in Metamath s reverse Polish nota tion for proofs a previous step can be referred to only once The repetition of step 1 here will enable you to see more clearly the correspondence of this proof with the Metamath version on p pah Our theorem is more properly called a theorem scheme for it repre sents an infinite number of theorems one for each possible term t Two examples of actual theorems would be 0 0 and 0 0 0 0 Rarely do we prove actual theorems since by proving schemes we can prove an infinite number of theorems in one fell swoop Similarly our proof should really be called a proof scheme To obtain an actual proof pick an actual te
92. Section 4 2 8 In ordinary mathematics formulas may arise that are true if the vari ables in them are distinct but become false when those variables are made identical For example the formula in logic 3x x 4 y which means for a given y there exists an x that is not equal to y is true in most mathemat ical theories namely all non trivial theories i e those that describe more than one individual such as arithmetic However if we substitute y with x we obtain Jz x x which is always false as it means there exists some thing that is not equal to itself P The d statement allows you to specify 9If you are a logician you will recognize this as the improper substitution of a free variable with a bound variable Metamath makes no inherent distinction between free and bound variables instead you let Metamath know what substitutions are permissible by using d statements in the right way in your axiom system 4 2 THE BASIC KEYWORDS 101 a restriction that forbids the substitution of one variable with another In this case we would use the statement d xy to specify this restriction The order in which the variables appear in a d statement is not impor tant We could also use d y x The d statement is actually more general than this as the disjoint in its name suggests The full meaning is that if any substitution is made to its two variables during the course of a proof that references
93. The set EXc corresponds to the set of permissible expressions for e a and p statements The set EX2 corresponds to the set of permissible expressions for f statements We denote by V e the set of all variables in an expression e EX i e the set of all a VR such that a en for some n lt e We also denote with abuse of notation by V the set of all variables in a collection of expressions E C EX i e Uecg V e C 2 3 Substitution Given a function F from VR to EX we denote by or or just the function from EX to EX defined recursively for nonempty sequences by o lt a gt F a for a VR a lt a gt lt a gt for a VR a g h a g o h for g h EX We also define o We call o a simultaneous substitution or just substitution with substitution map F We also denote with abuse of notation by o a substitution on a collection of expressions C EX i e the set a e e E The collection o may of course contain fewer expressions than because duplicate expressions could result from the substitution C 2 4 Statements We denote by DV the set of all unordered pairs a 8 C VR such that a B DV stands for distinct variables 164 APPENDIX C METAMATH S FORMAL SYSTEM A pre statement is a quadruple D T H A such that D C DV T C EX H C EXc and H is finite A EXc V H U A C V T and Ve f T V e 4 V f or equivalently e1 f whenever e f The
94. abase into a web page file called mmtheorems html When additional files are needed they are called mmtheorems2 html mmtheorems3 htm1l etc Optional command qualifier theorems_per_page number This qualifier specifies the number of statements to write per web page The default is 100 Note In version 0 07 30 of Metamath the Related Theorems links on the individual web pages presuppose 100 theorems per page when linking to the theorem list pages Therefore the theorems_per_page qualifier if it specifies a number other than 100 will cause the individual web pages to be out of sync and should not be used to generate the main theorem list for the web site This is expected to be fixed in a future version 5 9 TEXT FILE UTILITIES 151 5 8 3 write bibliography Command Syntax write bibliography filename This command reads an existing HTML bibliographic cross reference file normally called mmbiblio html and updates it per the bibliographic links in the database comments The file is updated between the HTML comment lines lt START gt and lt END gt The original input file is renamed to filename 1 A bibliographic reference is indicated with the reference name in brack ets such as Theorem 3 1 of Monk p 22 See Section 4 4 2 p 121 for syntax details 5 8 4 write recent_additions Command Syntax write recent_additions filename limit number This command reads an existing Recent Additions
95. abel of the first axiomatic assertion is tze A f statement must contain exactly two math symbols a constant followed by a variable The e a and p statements each start with a constant followed by in general an arbitrary sequence of math symbols Associated with each assertion is a set of hypotheses that must be sat isfied in order for the assertion to be used in a proof These are called the mandatory hypotheses of the assertion Among those hypotheses whose scope described below includes the assertion e hypotheses are always mandatory and f hypotheses are mandatory when they share their vari able with the assertion or its e hypotheses The exact rules for determining which hypotheses are mandatory are described in detail in Sections 4 2 7 and 4 2 8 For example the mandatory hypotheses of assertion tpl are tt and tr whereas assertion tze has no mandatory hypotheses because it contains no variables and has no e hypothesis Metamath s show statement com mand described in the next section will show you a statement s mandatory hypotheses Sometimes we need to make a hypothesis relevant to only certain asser tions The set of statements to which a hypothesis is relevant is called its scope The Metamath brackets and define a block that delimits the scope of any hypothesis contained between them The assertion mp has mandatory hypotheses wp wq min and maj The only mandatory hypoth esis of thi
96. ables it contains are ph ps ch th MM gt show proof prth essential lemmon renumber 1 pm3 2 p l ps gt th gt ps th 2 1 syl3dt p l ps gt ch gt th gt ch gt ps th 3 2 syl3 p l ph gt ps gt ph gt ch gt th gt ch gt ps A th 4 3 com23 p l ph gt ps gt ch gt th gt ph gt ch gt ps th 5 4 impa p l ph gt ps ch gt th gt ph gt ch gt ps th 6 impexp p l ph ch gt ps th lt gt ph gt ch gt ps th 7 5 6 sylibr p l ph gt ps ch gt th gt ph ch gt ps th There are references to a lot of unfamiliar statements To see what they are you may type the following 88 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED MM gt show proof prth statement_summary essential Summary of statements used in the proof of prth Statement syl3 is located on line 355 of the file set mm Inference adding common antecedents in an implication syl3 1 e ph gt ps syl3 p l ch gt ph gt ch gt ps Statement syl3dt is located on line 438 of the file set mm Deduction adding nested antecedents syl3dt 1 e ph gt ps gt ch syl3dt p I ph gt th gt ps gt th gt ch etc Of course you can look at e
97. ach of these statements and their proofs and so on back to the axioms of propositional calculus if you wish The search command is useful for finding statements when you know all or part of their contents The following example finds all statements con taining ph gt ps followed by ch gt th The is a wildcard that matches anything the before the prevents conflicts with math symbol token names The after SEARCH is also a wildcard that in this case means match any label MM gt SEARCH ph gt ps ch gt th 1521 prth p l ph gt ps ch gt th gt ph ch gt ps th 1522 pm3 48 p lI ph gt ps ch gt th gt ph ch J gt ps V th 1739 elimant p l ph gt ps ps gt ch gt ph gt th gt ph gt ch gt th Three statements prth pm3 48 and elimant were found to match To see what axioms and definitions prth ultimately depends on for its proof you can have the program backtrack through the hierarchy of theo rems and definitions MM gt show trace_back prth essential axioms Statement prth assumes the following axioms a statements ax 1 ax 2 ax 3 ax mp df bi df an Note that the 3 axioms of propositional calculus and the modus ponens rule are needed as expected in addition there are a couple of definitions 3 8 EXPLORING THE SET THEORY DATABASE 89 that are used along the w
98. acquired through exercises and hard work being able to see how gaps in a proof are filled in can give you increased confidence that can speed up the learning process and save you time when you get stuck The Metamath language breaks down a mathematical proof into its tini est possible parts These can be pieced together like interlocking pieces in a puzzle only in a way that produces correct and absolutely rigorous mathematics The nature of Metamath enforces very precise mathematical thinking similar to that involved in writing a computer program A crucial difference though is that once a proof is verified by the Metamath program to be correct it is definitely correct it can never have a hidden bug After getting used to the kind of rigor and accuracy provided by Metamath you might even be tempted to adopt the attitude that a proof should never be considered correct until it has been verified by a computer just as you would not completely trust a manual calculation until you have verified it on a calculator My goal for Metamath was a system for describing and verifying math ematics that is completely universal yet conceptually as simple as possible In approaching mathematics from an axiomatic formal viewpoint I wanted Metamath to be able to handle almost any mathematical system not nec essarily with ease but at least in principle and hopefully in practice I wanted it to verify proofs with absolute rigor and for this reas
99. actical purpose You decide to seek a quicker path You figure you ll just trace your way back to the beginning step by step as you would do with a computer program until you understand it But you quickly find that this is not possible since you can t even understand enough to know what you have to trace back to Maybe a different approach is in order maybe you should start at the beginning and work your way up First you read the introduction to the book to find out what the prerequisites are In a similar fashion you trace your way back through two or three more books finally arriving at one that seems to start at a beginning it lists the axioms of arithmetic Aha you naively think This must be the starting point the source of all mathemat ical knowledge Or at least the starting point for mathematics dealing with numbers you have to start somewhere and have no idea what the starting point for other mathematics would be But the word axioms looks promis ing So you eagerly read along and work through some elementary exercises at the beginning of the book You feel vaguely bothered these don t seem like axioms at all at least not in the sense that you want to think of axioms Axioms imply a starting point from which everything else can be built up according to precise rules specified in the axiom system Even though you can understand first few proofs in an informal way and are able to do some of the exercises
100. ailing and a left paren thesis will be removed The author notation will also create an entry in the bibliography cross reference file generated by write bibliography Section 5 8 3 For this to work properly the surrounding comment must be formatted as follows keyword label noise word author p number for example Theorem 5 2 of Monk p 223 The keyword is not case sensitive and must be one of the following theorem lemma definition compare proposition corollary axiom rule remark exercise problem notation example property figure postulate equation scheme chapter The optional label may consist of more than one non keyword and non noise word word The optional noise word is one of of in from on and is ignored when the cross reference file is created The write biblio graphy command will perform error checking to verify the above format A good way to become familiar with the markup notation is to look at the extensive examples in the set mm database 4 4 3 Including Other Files in a Metamath Source File The keywords and specify a file to be included at that point in a Metamath source file The syntax for including a file is as follows I file name The file name should be a single token with the same syntax as a math symbol i e all 93 printable characters other than are allowed subject to the file naming limitations of your operating system Comments may appear between the and keywords Incl
101. ain text Poincar conjecture Polish notation pop postfix connective power class power set 72 INDEX pre statement in a formal system 164 predicate calculus prefix connective Principia Mathematica printable character printers proof compressed 124 Lemmon style Metamath Metamath description of normal tree style HT Proof Assistant ix 46H48 5 ma 119 proof length 1 proof scheme 87 proper class 70 proper definition 108 proper substitution 100 170 propositional calculus provable assertion provable statement in a formal system 165 Purinton Josh push QED project qualifying expression quantifier theory quantum logic quantum mechanics Quine Willard Van Orman R go Eduardo range rational number 193 real and complex numbers axioms for real number recursion operator 127 127 recursive definition 112 redeclaration of symbols 39 imps reduct in a formal system 164 reflection principle relation restriction reverse Polish notation RPN 115 Robbins algebra Robbins Herbert Robinson s resolution principle BI Robinson T ae Rourke Colin RPN order 42 Fa my Eo RPN stack 40 95 115 178 Rucker Rudy ates rule fi 17 31 35 59 108 rule of generalization 61 Russell s paradox 9 Russell Bertrand Schr der Bernstein theorem scope 5 00 scoping statement s
102. an axiom then discovered that it could be derived as a theorem from the other axioms Because of this it was not necessary to keep it as an axiom By eliminating it the final set of axioms became that much simpler Unless you have worked with formal proofs before it probably wasn t apparent to you that t t could be derived from our two axioms until you saw the proof While you certainly believe that t t is true you might not be able to convince an imaginary skeptic who believes only in our two axioms until you produce the proof Formal proofs such as this are hard to come up with when you first start working with them but after you get used to them they can become interesting and fun Once you understand the idea behind formal proofs you will have grasped the fundamental principle that underlies all of mathematics As the mathematics becomes more sophisticated its proofs become more challenging but ultimately they all can be broken down into individual steps as simple as the ones in our proof above Mendelson s book from which our example was taken contains a number of detailed formal proofs such as these and you may be interested in looking it up The book is intended for mathematicians however and most of it is rather advanced Popular literature describing formal proofs include p 296 and 24 pp 204 230 2 2 2 Converting It to Metamath Formal proofs such as the one in our example break down logical reasoning into small p
103. and as far as it is concerned wff is just an arbitrary constant symbol in a math symbol sequence The mandatory hypotheses of wnew are wp wr and ws wq is an optional hypothesis In our particular proof the optional hypothesis is not referenced but in general any combination of active i e optional and mandatory hypotheses could be referenced The proof of statement wnew is the sequence of five labels starting with ws step 1 and ending with w2 step 5 When Metamath verifies the proof it scans the proof from left to right We will examine what happens at each step of the proof The stack starts off empty At step 1 Metamath looks up label ws and determines that it is a hypothesis so it pushes the symbol sequence of statement ws onto the stack Stack location Contents 1 wff s Metamath sees that the labels wr and wp in steps 2 and 3 are also hy potheses so it pushes them onto the stack After step 3 the stack looks like this Stack location Contents 3 wff p 2 wff r 1 wff s At step 4 Metamath sees that label w2 is an assertion so it must do some processing First it associates the mandatory hypotheses of w2 which are wp and wq with stack locations 2 and 3 in that order Metamath determines that the only possible way to make hypothesis wp match become identical 4 3 THE ANATOMY OF A PROOF 117 to stack location 2 and wq match stack location 3 is to substitute variabl
104. and outside of math mode and will not cause the output processor to enter or exit math mode Outside of math mode any token preceded by a tilde will be format ted in typewriter font and the tilde removed to make them stand it from the rest of the text This formatting will be applied from all characters after the tilde up to the first white space This formatting mode is called label 18Computer languages have differing standards for nested comments and rather than picking one it was felt simplest not to allow them at all at least in the current version of Metamath 0 07 30 120 CHAPTER 4 THE METAMATH LANGUAGE mode If a literal tilde is desired outside of math mode use two tildes in a row to represent it These markup features have to do only with how the comments are type set and have no effect on how Metamath verifies the database The improper use of them may result in incorrectly typeset output but no Metamath error messages will result during the read and verify proof commands How ever the new write theorem_list command added in version 0 07 30 of the Metamath program will check for markup errors as a side effect of its HTML generation Section 5 7 has instructions for creating TFX output Math Symbols In Comments The grave accent tells Metamath to switch a comment to math mode In this mode the characters following the are interpreted as a sequence of math symbol tokens separated by white space
105. arate disk and folder names from your file name For example volume file name refers to the root di rectory volume folder name file name refers to a folder in root and volume folder name file name refers to a deeper folder A simple file name refers to a file in the folder from which you launch the Metamath application Under Mac OS X and later the Metamath program is run under the Terminal application which conforms to Unix naming conventions 124 CHAPTER 4 THE METAMATH LANGUAGE label list compressed proof The first serves as a flag to Metamath that a compressed proof follows The label list includes all statements referred to by the proof except the mandatory hypotheses The compressed proof is a compact encoding of the proof using upper case letters and can be thought of as a large integer in base 26 White space inside of compressed proof is optional and is ignored It is important to note that the order of the mandatory hypotheses of the statement being proved must not be changed if the compressed proof format is used otherwise the proof will become incorrect The reason for this is that the mandatory hypotheses are not mentioned explicitly in the compressed proof in order to make the compression more efficient If you wish to change the order of mandatory hypotheses you must first convert the proof back to normal format using the save proof statement normal command Later you can go back to compressed format
106. are inherently hard to follow to begin with and few people enjoy verifying them manually in detail 30A wf or well formed formula is a mathematical expression string of symbols con structed according to some precise rules A formal mathematical system contains 1 the rules for constructing syntactically correct wffs 2 a list of starting wffs called axioms and 3 one or more rules prescribing how to derive new wffs called theorems from the axioms or previously derived theorems An example of such a system is contained in Metamath s set theory database which defines a formal system from which all of stan dard mathematics can be derived Section 2 2 1 steps you through a complete example of a formal system and you may want to skim it now if you are unfamiliar with the concept 311t is possible of course to specify wff construction syntax outside of the program itself with a suitable input language the Metamath language being an example but some proof verification or theorem proving programs lack the ability extend wff syntax in such a fashion 20 CHAPTER 1 INTRODUCTION Metamath takes a different approach Metamath s knowledge is lim ited to the ability to substitute variables for expressions subject to some simple constraints Once the basic algorithm of Metamath is assumed to be debugged and perhaps independently confirmed it can be trusted once and for all The information that Metamath needs to understand m
107. are often more practical to use than Metamath but they can have a steep learning curve for those without a mathematical background For example one usually must have a fair understanding of mathematical logic in order to follow their proofs For the working mathematician Mizar is an excellent tool for rigorously documenting proofs Mizar typesets its proofs in the informal English used by mathematicians and while fine for them are just as inscrutable by laypersons A price paid for Mizar is a relatively steep learning curve of a couple of weeks Several mathematicians are actively formalizing different areas of mathematics using Mizar and publishing the proofs in a dedicated journal Unfortunately the task of formalizing mathematics is still looked down upon to a certain extent since it doesn t involve the creation of new mathematics To summarize our discussions of computers and mathematics computer algebra systems can be viewed as theorem generators focusing on a narrow realm of mathematics numbers and their properties automated theorem provers as proof generators for specific theorems in a much broader realm covered by a built in formal system such as first order logic proof verifiers in general as proof documentors usually restricted to first order logic and Metamath in particular as a proof documentor whose realm is essentially unlimited 1 4 MATHEMATICS AND METAMATH 29 1 4 Mathematics and Metamath 1 4 1 Standard Mathematic
108. ariables x and y are equal Let s look at what happens if we edit the database to comment out this requirement d xz d yz Theorem to add distinct quantifier to atomic formula axi7eq p l x y gt A z2x y When it tries to verify the proof Metamath will tell you that x and z must be disjoint because one of its steps references an axiom or theorem that has this requirement MM gt verify proof axi7eq axi7eq Error at statement 1918 label axi7eq type p vz wal wi vx vy vz ax 12 vx vy weq vz vx ax 16 vx vy There is a disjoint variable d violation at proof step 29 Assertion ax 16 requires that variables x and y be disjoint But x was substituted with z and y was substituted with x The assertion being proved axi7eq does not require that variables z and x be disjoint We can see the substitutions into ax 16 with the following command MM gt show proof axi7eq detailed_step 29 Proof step 29 pm2 61dd 2 ax 16 a A zz x gt x y gt A zx y This step assigns source ax 16 a to target pm2 61dd 2 104 CHAPTER 4 THE METAMATH LANGUAGE e The source assertion requires the hypotheses wph f step 26 vx f step 27 and vy f step 28 The parent assertion of the target hypothesis is pm2 61dd p step 36 The source assertion before substitution was ax 16 a A x x y gt ph gt A x ph The following substitutions w
109. as built So that you can repeat these steps we will first have the Proof Assistant erase the proof in Metamath s source buffer then reconstruct it The source buffer is the place in memory where Metamath stores the information in the database when it is read in New or modified proofs are kept in the source buffer until a write source command is issued In practice you would place a between and in the database to indicate to Metamath that the proof is unknown and that would be your starting point Whenever verify proof command encounters a proof with a in place of a proof step the statement is identified as not proved Before you start you should write down on a piece of paper the complete formal proof as it would appear with the essential qualifier in a show proof command see the display of show proof thi essential lemmon 2 4 YOUR FIRST PROOF 47 renumber above as an example After you get used to using the Proof Assistant you may get to a point where you can see the proof in your mind and let the Proof Assistant guide you in filling in the details at least for simpler proofs but until you gain that experience it is important to write down all the details in advance Otherwise you may waste a lot of time as you let it take you down a wrong path A proof is developed with the Proof Assistant by working backwards starting with the theorem to be proved and assigning each unknown step with a theorem or hypothesis un
110. asp the concepts needed for deep understanding Metamath allows you to approach the proof of the theorem from a quite different per spective peeling apart the formulas and definitions layer by layer until an entirely different kind of understanding is achieved Every step of the proof is there pieced together with absolute precision and instantly available for inspection through a microscope with a magnification as powerful as you desire A proof in itself can be considered an object of beauty Constructing an elegant proof is an art Once a famous theorem has been proved of ten considerable effort is made to find simpler and more easily understood proofs Creating and communicating elegant proofs is a major concern of mathematicians Metamath is one way of providing a common language for archiving and preserving this information The length of a proof can to a certain extent be considered an objective measure of its beauty since shorter proofs are usually considered more elegant In the set theory database set mm provided with Metamath one goal was to make all proofs as short as possible 1 1 5 Simplicity God made man simple man s complex problems are of his own devising Eccues 7 284 God made integers all else is the work of man LEOPOLD KRONECKER2 For what is clear and easily comprehended attracts the compli cated repels DAVID HILBERT The Metamath language is simple and Spartan Metamath treats all
111. assertion s frame may be substituted with whatever is needed in a proof step that references the assertion whereas constants remain fixed and may not be substituted with anything The particular token used for a variable in an assertion s frame is irrelevant when the assertion is referenced in a proof and it doesn t matter if that token is not available outside of the referenced assertion s frame Constants however must be globally fixed 14To keep things slightly simpler we do not bother to define the concept of active for the scoping statements 114 CHAPTER 4 THE METAMATH LANGUAGE In the present version of the Metamath language there is no theoretical benefit for the feature allowing variables to be active for limited scopes rather than global It is just a convenience that allows them for example to be locally grouped together with their corresponding f variable type declarations E5 Frames Revisited Now that we have covered scoping we will look at how an arbitrary Meta math database can be converted to the simple sequence of extended frames described on p 111 This is also how Metamath stores the database in ternally when it reads in the database source The method is simple First we collect all constant and variable c and v declarations in the database ignoring duplicate declarations of the same variable in different scopes We then put our collected c and v declarations at the beginning of the d
112. at end with The Metamath tokens c and v each declare a list of user defined tokens called math symbols that the database will reference later on All of the math symbols they define you have seen earlier except the turnstile symbol l F which is commonly used by logicians to mean a proof exists for For us the turnstile is just a convenient symbol that distinguishes expressions that are axioms or theorems from expressions that are terms or wffs The c statement declares constants and the v statement declares 2 2 YOUR FIRST FORMAL SYSTEM 39 variables or more precisely metavariables A variable may be substi tuted with sequences of math symbols whereas a constant may not be sub stituted with anything It may seem redundant to require both c and v statements since any math symbol not specified with a c statement could be presumed to be a variable but this provides for better error checking and also allows math symbols to be redeclared Section 4 2 8 The token f specifies a statement called a variable type hypothesis also called a floating hypothesis and e specifies a logical hypothesis also called an essential hypothesis The token a specifies an axiomatic assertion and p specifies a provable assertion To the left of each oc currence of these four tokens is a label that identifies the hypothesis or assertion for later reference For example the l
113. atabase so that their scope is the entire database Next for each assertion in the database we determine its frame and extended frame The extended frame is simply the f e and d statements that are active The frame is the extended frame with all optional hypotheses removed An equivalent way of saying this is that the extended frame of an asser tion is the collection of all f e and d statements whose scope includes the assertion in the order they appear 4 3 The Anatomy of a Proof Each provable assertion p statement in a database must include a proof The proof is located between the and keywords in the p statement In the basic Metamath language a proof is a sequence of statement la bels This label sequence serves as a set of instructions that the Metamath program uses to construct a series of math symbol sequences The construc tion must ultimately result in the math symbol sequence contained between the p and keywords of the p statement Otherwise the Metamath program will consider the proof incorrect and it will notify you with an appropriate error message when you ask it to verify the proof 6 Each label in a proof is said to reference its corresponding statement Associated with any assertion p or a statement is a set of hypotheses f or e statements that are active with respect to that assertion Some are 15A future version of the Metamath language may extend the language to include true loca
114. athematics is contained entirely in the body of knowledge presented to Metamath Any errors in reasoning can only be errors in the axioms or definitions contained in this body of knowledge As a constructive language Metamath has no conditional branches or loops like the ones that make computer programs hard to decipher instead the language can only build new sequences of symbols from earlier sequences of symbols The simplicity of the rules that underlie Metamath not only makes Meta math easy to learn but also gives Metamath a great deal of flexibility For example Metamath is not limited to describing standard first order logic higher order logics and fragments of logic can be described just as easily Metamath gives you the freedom to define whatever wff notation you prefer it has no built in conception of the syntax of a wff With suitable axioms and definitions Metamath can even describe and prove things about itself John Harrison discusses the reflection principle involved in self descriptive sys tems in 20 The flexibility of Metamath requires that its proofs specify a lot of detail much more than in an ordinary formal proof For example in an ordinary formal proof a single step consists of displaying the wff that constitutes that step In order for a computer program to verify that the step is acceptable it first must verify that the symbol sequence being displayed is an acceptable wff Most proof verifiers hav
115. aw a picture illustrating the concept for example a circle containing dots could represent a collection of sets and a smaller circle drawn inside of the circle could represent a subset Overlapping circles can illustrate intersection and union Circles that illus trate the concepts of set theory are frequently used in elementary textbooks and are called Venn diagrams 1 Axiom of Extensionality Two sets are identical if they contain the same elements 2 Axiom of Pairing The set x y exists 3 Axiom of Power Sets The power set of a set the collection of all of its subsets exists For example the power set of x y is 2 x y x y and it exists 4 Axiom of the Null Set The empty set exists 5 Axiom of Union The union of a set the set containing the elements of its members exists For example the union of x y z is x y z and it exists 6 Axiom of Regularity Roughly no set can contain itself nor can there be membership loops such as a set being an element of one of its members 7 Axiom of Infinity An infinite set exists An example of an infinite set is the set of all integers 8 Axiom of Separation The set exists that is obtained by restricting x with some property For example if the set of all integers exists then the set of all even integers exists 9 Axiom of Replacement The range of a function whose domain is restricted to the elements of a set x is also a set For example there
116. axioms require a higher level of sophistication they can be more practical to deal with on an everyday informal basis Even if you are just developing Metamath proofs familiarity with the traditional approach can help you arrive at a proof outline much faster which you can then convert to the detail required by Metamath There is also a new rule of inference in predicate calculus if y is an axiom or a theorem then Vx is also a theorem This is called the rule of generalization Unlike propositional calculus no decision procedure analogous to the truth table method exists nor theoretically can exist that will definitely determine whether a formula is a theorem of predicate calculus Much of the work in the field of automated theorem proving has been dedicated to coming up with clever heuristics for proving theorems of predicate calculus but they can never be guaranteed to work always 3 2 3 Equality Predicate calculus may be presented with or without axioms for equality We will require the axioms of equality as a prerequisite for the version of set theory we will use In standard systems the axioms for equality are the following two LT T xz y gt y x x p x y where p x y arises from y x x by replacing some but not necessarily all free occurrences of x by y provided that y is free for x in x x The first equality axiom is simple but the condition on the second one is somewhat awkward to impl
117. ay Note that Metamath makes no distinction between axioms and definitions In set mm they have been distinguished artificially by prefixing their labels with ax and df respectively For ex ample df an defines conjunction logical AND which is represented by the symbol Section 4 5 discusses the philosophy of definitions and the Meta math language takes a particularly simple conservative approach by using the a statement for both axioms and definitions You can also have the program compute how many steps a proof has if we were to follow it all the way back to a statements MM gt show trace_back prth essential count_steps The statement s actual proof has 5 steps Backtracking a total of 55 different subtheorems are used The statement and subtheorems have a total of 196 actual steps If subtheorems used only once were eliminated there would be a total of 25 subtheorems and the statement and subtheorems would have a total of 143 steps The maximum path length is 20 A longest path is prth lt imp4b lt imp4a lt impexp lt imbili lt impbi lt bi3 lt expi lt expt lt pm3 2im lt con2d lt con2 lt nega lt pm2 18 lt pm2 43i lt pm2 43 lt pm2 27 lt comi2 lt syl lt ali lt ali 1 This tells us that we would have to inspect 196 steps if we want to verify the proof completely starting from the axioms A few more statistics are also shown There are one or more paths back to axioms tha
118. bjects at all but deals directly with metavariables as its primitive objects from the start This is a subtle but key distinction you should keep in mind and it makes our definition of formal system somewhat different from that typically found in the literature So the a 3 etc above are actually metametavariables when used to represent vg v1 This section is taken mostly verbatim from Tarski 61 p 63 C 2 THE FORMAL DESCRIPTION 163 Finite sequences all terms of which are symbols are called expressions EX is the set of all expressions thus EX U n SM nEw A constant prefixed expression is a an expression of non zero length whose first term is a constant We denote the set of all constant prefixed expres sions by EXc e EX e gt 0 eo E CN A constant variable pair is an expression of length 2 whose first term is a constant and whose second term is a variable We denote the set of all constant variable pairs by EX2 e EXc e 2Ae1 VR Relationship to Metamath In general the set SM corresponds to the set of declared math symbols in a Metamath database the set CN to those declared with c statements and the set VR to those declared with v statements Of course a Metamath database can only have a finite number of math symbols whereas formal systems in general can have an infinite number although the number of Metamath math symbols available is in principle unlimited
119. bstract mathematical objects that obey a certain surprisingly simple set of axioms See for example Munkres 41 On the other hand writing proofs in the Metamath language is challenging requiring a degree of rigor far in excess of that normally taught to students In a classroom setting I doubt that writing Metamath proofs would ever replace traditional homework exercises involving informal proofs because the time needed to work out the details would not allow a course to cover much material For students who have trouble grasping the implied rigor in traditional material writing a few simple proofs in the Metamath language might help clarify fuzzy thought processes Although somewhat difficult at first it eventually becomes fun to do like solving a puzzle because of the instant feedback provided by the computer 4 CHAPTER 1 INTRODUCTION Metamath is probably unlike anything you have encountered before In this first chapter we will look at the philosophy and use of computers in mathematics in order to better understand the motivation behind Meta math The material in this chapter is not required in order to use Meta math You may skip it if you are impatient but I hope you will find it educational and enjoyable If you want to start experimenting with the Metamath program right away proceed directly to Chapter 2 p 33 To learn the Metamath language skim Chapter P then proceed to Chapter p P1 1 1 Mathematics as a Co
120. calculations in a second so maybe it s still practical to do Now the puzzle becomes how to figure out what the million steps are that each English language step corresponds to Your mathematician friend hasn t a clue but suggests that maybe you would find the answer by studying set theory Actually your friend thinks you re a little off the wall for even wondering such a thing For him this is not what mathematics is all about The subject of set theory keeps popping up so you decide it s time to look it up You decide to start off on a careful footing so you start reading a couple of very elementary books on set theory A lot of it seems pretty obvious like intersections subsets and Venn diagrams You thumb through one of the books nowhere is anything about axioms mentioned The other book relegates to an appendix a brief discussion that mentions a set of axioms called Zermelo Fraenkel set theory and states them in English You look at them and have no idea what they really mean or what you can do with them The comments in this appendix say that the purpose of mentioning them is to expose you to the idea but imply that they are not necessary for basic understanding and that they are really the subject matter of advanced treatments where fine points such as a certain paradox Russell s paradox are resolved Wait a minute shouldn t the axioms be a starting point not an ending point If there are paradoxes that arise wi
121. case of Definition 4 16 p 14 of Takeuti and Zaring Note that x y and z must be distinct but that x and y may occur in g df opab a F x y p z dedy z df opab a lt x y gt ph lt x y gt ph Define the epsilon relation Similar to Definition 6 22 p 30 of Takeuti and Zaring df eprel a H z y rey df eprel a E lt x y gt xe y x y Avp zl E xE y ze Define a founded relation R is a founded relation on A iff if and only if each nonempty subset of A has an R minimal element Similar to Definition 6 21 p 30 of Takeuti and Zaring df fr a F RFrAGVa t CAAnL 8 gt Ay yerdA an z z Ry 74 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED df fr a R Fr A lt gt Ape CG BN Re Se gt E y ye x N A xiid izilizRy 0 6 6 Define a well ordering R is a well ordering of A iff it is founded on A and the elements of A are pairwise R comparable Similar to Definition 6 24 2 p 30 of Takeuti and Zaring df we a k RWeAG RFr AnNYzYy rEA nyEA gt zRyvVv c yVyRex df we a R We A lt gt RFrA A x A yy CC xe A ye A gt CxRy V x y VyRx amp Define the ordinal predicate which is true for a class that is transitive and is well ordered by the epsilon relation Similar to definition on p 468 Bell and Machover df ord a Ord A Tr AAE We A df ord a Ord A lt gt Tr A EWeA Define class of a
122. ce each step can be verified automatically There are several other approaches besides Metamath to a project such as this Section 1 3 3 discusses some of these To me a noble goal would be a CD ROM with hundreds of thousands of theorems and their computer verifiable proofs encompassing a significant fraction of known mathematics and available for instant access Whether or not Metamath is an appropriate choice remains to be seen but in principle I believe it is sufficient PREFACE xiii Formalism Over the past fifty years a group of French mathematicians working collec tively under the pseudonym of Bourbaki have co authored a series of mono graphs that attempt to rigorously and consistently formalize large bodies of mathematics from foundations On the one hand certainly such an effort has its merits on the other hand the Bourbaki project has been criticized for its scholasticism and hyperaxiomatics that hide the intuitive steps that lead to the results 3 p 191 Metamath unabashedly carries this philosophy to its extreme and no doubt is subject to the same kind of criticism Nonetheless I think that in conjunction with conventional approaches to mathematics Metamath can serve a useful purpose The Bourbaki approach is essentially pedagogic re quiring the reader to become intimately familiar with each detail in a very large hierarchy before he or she can proceed to the next step The differ ence with Metamath is tha
123. ch command minimize_with command more command open log command open tex command 98 147 prove command 47 read command 46 83 134 save new_proof command 52 124 146 187 188 save proof command Pheer search command 84 88 eT set echo command 132 set empty_substitution command height command scroll command set search_limit command 142 set unification_timeout command set width command show trace_back command show labels command 42 show memory command show new_proof eg 474 142 show proof command 3 1 42 45 6 5 Bo 17 settings command 135 statement TENSE fa 42 show trace_back Ba ABs show usage command BO 137 submit naa Be 5 132 set set show show tools command unify command verify proof command 41 114 write bibliography command 151 write recent_additions command 151 write source command 46 134 write theorem_list command 150 abstract algebra 3 abstraction class fm 56 70 of nested ordered Aia of ordered pairs 73 active math symbol 93 100 113 active statement addition 127 of ordinals ambiguous unification 146 INDEX analysis Anderson Alan Ross Andr ka H Appel K ASCII 33 92 96 assertion 106 in a formal system 1 assertion label 40 115 Auden W H Aussonderane a automated proof verification 19 aaa theorem proving 21
124. d assume x and y are distinct variables y e The statement d xy should be read assume x does not occur in y p e The statement d x A should be read assume variable x does not occur in class A e The restriction and hypothesis group d ct A d cw e FK x A gt por is frequently used in place of explicit substitution meaning assume y results from the proper substitution of A for x in y Sometimes e F Y gt Yx y is used instead of d zy which requires only that x be effectively not free in y but not necessarily absent from it The use of implicit substitution is partly a matter of personal style although it may make proofs somewhat shorter than would be the case with explicit substitution e The hypothesis e FAEV should be read assume class A is a set i e exists This is a conve nient convention used by Quine e The restriction and hypothesis 78 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED d xy e ye A gt Va yEA should be read assume variable x is effectively not free in class A 3 6 A Theorem Sampler In this section we list some of the more important theorems that are proved in the set mm database and they illustrate the kinds of things that can be done with Metamath While all of these facts are well known results in set theory Metamath offers the advantage of easily allowing you to trace their derivation back to axioms Our intent here is not to
125. d Metamath s proof notation and Metamath makes use of a very simple version of it Section 4 3 1 1 4 5 Metamath and First Order Logic First order logic is the supporting structure for standard mathematics On top of it is set theory which contains the axioms from which virtually all of mathematics can be derived a remarkable fact 4 One of the things that makes Metamath more practical for first order theories is a set of axioms for first order logic designed specifically with Metamath s approach in mind These are included in a standard database called set mm which comes with the Metamath software See Chapter for a detailed description the axioms are shown in Section 3 3 While logi cally equivalent to standard axiom systems our axiom system breaks up the standard axioms into smaller pieces such that from them you can directly derive what in other systems can only be derived as higher level metathe 44 An exception seems to be category theory There are several schools of thought on whether category theory is derivable from set theory At a minimum it appears that an additional axiom is needed that asserts the existence of an inaccessible cardinal a type of infinity so large that standard set theory can t prove or deny that it exists For more information see pp 328 331 and 6 32 CHAPTER 1 INTRODUCTION orems In other words it is more powerful than the standard axioms from a metalogical point of vie
126. d in the compressed format which re duces storage requirements for a database See Appendix B 5 6 Creating Proofs Before using the Proof Assistant you must add a p to your source file using a text editor containing the statement you want to prove Its proof should consist of a single meaning unknown step Example equid p x x To enter the Proof assistant type prove label e g prove equid Meta math will respond with the MM PA gt prompt Proofs are created working backwards from the statement being proved primarily using a series of assign commands A proof is complete when all steps are assigned to statements and all steps are unified and completely known During the creation of a proof Metamath will allow only operations that are legal based on what is known up to that point For example it will not allow an assign of a statement that cannot be unified with the unknown proof step being assigned Important You should figure out your first few proofs completely and write them down by hand before using the Proof Assistant Otherwise you will become extremely frustrated The Proof Assistant is not a tool to help you discover proofs It is just a tool to help you add them to the database For a tutorial read Section 2 4 To practice using the Proof Assistant you may want to prove an existing theorem then delete all steps with delete all then re create it with the Proof Assistant while looking at its proof dis
127. d the Axiom of Separation can be derived from the other axioms and are therefore unnecessary although they tend to be included in standard texts for various reasons historical philosophical and possibly because some authors may not know this In the Metamath set theory database these redundant axioms are derived from the other ones 3 3 The Axioms in the Metamath Language The standard textbook axioms of predicate calculus are somewhat cumber some to implement on a computer because of the complex notions of free variable and proper substitution While it is possible to use the Meta math language to implement these concepts we have chosen not to do so in the set mm set theory database Instead we have eliminated them by carefully crafting the axioms so as to avoid them This makes it easy for a beginner to follow the steps in a proof without knowing any advanced con cepts other than the simple concept of replacing variables with expressions Here we list the axioms as they appear in set mm so you can look them up there easily Incidentally the show statement tex command was used to typeset them In order to develop the concepts of free variable and proper substitution from the axioms described below we use an additional Metamath statement type called disjoint variable restriction that we have not encountered be fore In the context of the axioms the statement d xy simply means that x and y must be distinct i e the
128. de of a definition involves an individual variable not in the expression being defined it is done so that the right hand side remains independent of the particular dummy variable we use We continue to add to T the following definitions i e the reducts of the following pre statements for empty set class union and unordered pair They should be self explanatory Analogous to our use of 4 to define new wffs in Example 4 we use to define new abstraction terms and both may be read informally as is defined as in this context T class Y T x vr x T class AU B x A x B T H AU B a x AV ze B T class A B x A x B T F A B a z AV x B BN ok eX Ee ee C 4 Metamath as a Formal System This section presupposes a familiarity with the Metamath computer lan guage Our theory describes formal systems and their universes The Metamath language provides a way of representing these set theoretical objects to a 6Note that this definition makes unnecessary the introduction of a separate notation similar to y z y for proper substitution although we may choose to do so to be con ventional Incidentally y z y as it stands would be ambiguous in the formal systems of our examples since we wouldn t know whether 7 y z y meant y a y or gt y aly Instead we would have to use an unambiguous variant s
129. e We will conclude this quick introduction to Metamath with some helpful hints on how to navigate your way through the commands When you type commands into Metamath s CLI you only have to type as many characters of a command keyword as are needed to make it un ambiguous If you type too few characters Metamath will tell you what the choices are In the case of the read command only the r is needed to specify it unambiguously so you could have typed MM gt r demoO mm instead of MM gt read demoO mm In our description we always show the full command words When using the Metamath CLI commands in a command file to be read with the submit command it is good practice to use the unabbreviated command to ensure your instructions will not become ambiguous if more commands are added to the Metamath program in the future The command keywords are not case sensitive you may type either read or ReAd File names may or may not be case sensitive depending on your computer s operating system Metamath label and math symbol tokens are case sensitive The help command will provide you with a list of topics you can get help on You can then type help topic to get help on that topic If you are uncertain of a command s spelling just type as many char acters as you remember of the command If you have not typed enough characters to specify it unambiguously Metamath will tell you what choices you have MM gt show s Ambiguous keyword
130. e nature of definitions in the most general sense does not permit them to automatically be verified as proper the judgment of the mathematician is required to ensure it In set mm effort was made to make almost all def initions directly eliminable and thus minimize the need for such judgment If you are not a mathematician it may be best not to add or change any a statements but instead use the mathematical language already provided in standard databases This way Metamath will not allow you to make a mistake i e prove a false result 4 2 7 Frames We now introduce the concept of a collection of related Metamath statements called a frame Every assertion a or p statement in the database has an associated frame A frame is a sequence of d f and e statements zero or more of each followed by one a or p statement subject to certain conditions we will describe For simplicity we will assume that all math symbol tokens used are declared at the beginning of the database with c and v statements which are not properly part of a frame Also for simplicity we will assume there are only simple d statements those with only two variables and imagine any compound d statements those with more than two variables as broken up into simple ones A frame groups together those hypotheses and d statements relevant to an assertion a or p statement The statements in a frame may or may 4 2 THE BASIC KEYWORDS 109 not be
131. e p with r and q with p Metamath makes these substitutions into w2 and obtains the symbol sequence wff r gt p It removes the hypotheses from stack locations 2 and 3 then places the result into stack location 2 Stack location Contents 2 uff r gt p 1 wff s At step 5 Metamath sees that label w2 is an assertion so it must again do some processing First it matches the mandatory hypotheses of w2 which are wp and wq to stack locations 1 and 2 Metamath determines that the only possible way to make the hypotheses match is to substitute variable p with s and q with r gt p Metamath makes these substitutions into w2 and obtains the symbol sequence wff s gt r gt p It removes stack locations 1 and 2 then places the result into stack location 1 Stack location Contents 1 wif s gt r gt p After Metamath finishes processing the proof it checks to see that the contents of stack location 1 is the same as the math symbol sequence in the p statement This is the case for our proof of wnew so we have proved wnew successfully If the result differs Metamath will notify you with an error message An error message will also result if the stack contains more than one entry at the end of the proof or if the stack did not contain enough entries at any point in the proof to match all of the mandatory hypotheses of an assertion Finally Metamath will notify you
132. e A may be eliminated from a wff by replacing it with x where neither x nor y occur in the wf In this section individual variables are always assumed to be distinct from each other unless otherwise indicated In addition dummy variables on the right hand side of a definition do not occur in the class and wff metavariables in the definition The definitions we present here are a partial but self contained collection selected from several hundred that appear in the current set mm database They are adequate for a basic development of elementary set theory Define the abstraction class x and y need not be distinct Definition 2 1 of Quine p 16 This definition may seem puzzling since it is shorter than the 7Because of the way we initially defined wfts this is the case with any postfix connec tive one occurring after the symbols being connected or infix connective one occurring between the symbols being connected Metamath does not have a built in notion of operator binding strength that could eliminate the ambiguity The initial parenthesis effectively provides a prefix connective to eliminate ambiguity Some conventions such as Polish notation used in the 1930 s and 1940 s by Polish logicians use only prefix con nectives and thus allow the total elimination of parentheses at the expense of readability In Metamath we could actually redefine all notation to be Polish if we wanted to without having to change any proofs
133. e On Az rec w v v sucw z y Here the abstraction class of nested ordered pairs is defined by df oprab in set mm and rec is a recursion operator with the definition df rfg a Hrec F A U Ff 3xz xrEOnAa fFnzr Yylycesr gt f y g I 9 3A2 A V gt g VLimdomg Az F g Udomg LimdomgAz Urang fty which can be further broken down with definitions shown in Section You may be surprised at the complexity of what seems like such a simple notion From these definitions the simpler more intuitive recursive definition is derived as a set of theorems a V 128 CHAPTER 4 THE METAMATH LANGUAGE Chapter 5 The Metamath Program This chapter provides a reference manual for the Metamath program Current instructions for obtaining and installing the Metamath program can be found at the http metamath org web site For Windows there is a pre compiled version called metamath exe For Unix Linux and Mac OS X which we will refer to collectively as Unix the Metamath program can be compiled from its source code with the command gcc c o metamath using the gcc C compiler available on those systems In the command syntax descriptions below fields enclosed in square brackets are optional File names may be optionally enclosed in single or double quotes This is useful if the file name contains slashes such as in Unix path names that might be confused with Metamath command qualifi
134. e a hyperlink in the HTML output that links to the web page for statement label Exception if label begins with http it is assumed to be a URL which is used as is except that a in the URL should be specified with Only a and p statement labels may be used since web pages for e and f statements are not generated author Link to a bibliographical reference See help html and help write bibliography in the Metamath program for more infor mation See also Sections and t Flag the comment as the special one containing TeX and or HTML typesetting definitions See Section and Appendix A _ underscore Italicize text starting from space_non space i e _ with a space before it and a non space character after it until the next non space_space Normal punctuation e g a trailing comma or period is ignored when determining space _ underscore non space_non space string where non space string is a string of non space characters will make non space string become a subscript 122 CHAPTER 4 THE METAMATH LANGUAGE It is recommended that spaces surround any and tokens in the com ment and that a space follow the label after a token This will make global substitutions to change labels and symbol names much easier and also eliminate any future chance of ambiguity Spaces around these tokens are automatically removed in the final output to conform with normal rules of punctuation for example a space between a tr
135. e are advantages to documenting it in a standard format that will make it accessible to others Sixty years from now your field may be dormant and as Davis and Hersh put it your writings would become less translatable than those of the Maya I3 p 37 1 4 4 A History of the Approach Behind Metamath Probably the one work that has had the most motivating influence on Meta math is Whitehead and Russell s monumental Principia Mathematica 65 whose aim was to deduce all of mathematics from a small number of primi tive ideas in a very explicit way that in principle anyone could understand and follow While this work was tremendously influential in its time from a modern perspective it suffers from several drawbacks Both its notation and its underlying axioms are now considered dated and are no longer used From our point of view its development is not really as accessible as we would like to see for practical reasons proofs become more and more sketchy as its mathematics progresses and working them out in fine detail requires a degree of mathematical skill and patience that many people don t have There are numerous small errors which is understandable given the tedious 1 4 MATHEMATICS AND METAMATH 31 technical nature of the proofs and the lack of a computer to verify the de tails However even today Principia Mathematica stands out as the work closest in spirit to Metamath It remains a mind boggling work and one can t
136. e at least basic wff syntax built into their programs Metamath has no hard wired knowledge of what constitutes a wff built into it instead every wff must be explicitly constructed based on rules defining wffs that are present in a database Thus a single step in an ordinary formal proof may be correspond to many steps in a Metamath proof Despite the larger number of steps though this does not mean that a Metamath proof must be significantly larger than an ordinary formal proof The reason is that since we have constructed the wff from scratch we know what the wff is so there is no reason to display it We only need to refer to a sequence of statements that construct it In a sense the display of the wff in an ordinary formal proof is an implicit proof of its own validity as a wff Metamath just makes the proof explicit Section 4 3 describes Metamath s proof notation 1 2 Computers and Mathematicians The computer is important but not to mathematics 1 2 COMPUTERS AND MATHEMATICIANS 21 PAUL Hatmog2 Pure mathematicians have traditionally been indifferent to computers even to the point of disdain Computer science itself is sometimes considered to fall in the mundane realm of applied mathematics perhaps essential for the real world but intellectually unexciting to those who seek the deepest truths in mathematics Perhaps a reason for this attitude towards computers is that there is little or no computer software that meet
137. e courses from a competent mathematician and you ll catch on Student Are you sure I M Well it is possible that you won t if you don t have any aptitude for it That can happen too Student Then you decide what a proof is and if I don t learn to decide in the same way you decide I don t have any aptitude I M If not me then who THE IDEAL MATHEMATICIAN In the past century brilliant mathematicians have discovered almost unimaginably profound results that rank among the crowning intellectual achievements of mankind However there is a sense in which modern ab stract mathematics is behind the times stuck in an era before comput ers existed While no one disputes the remarkable results that have been achieved communicating these results in a precise way to the uninitiated is virtually impossible To describe these results a terse informal language is used which despite its elegance is very difficult to learn This informal language is not imprecise far from it but rather it often has omitted detail and symbols with hidden context that are implicitly understood by an ex pert but few others Extremely complex technical meanings are associated with innocent sounding English words such as compact and measurable that barely hint at what is actually being said Anyone who does not keep the precise technical meaning constantly in mind is bound to fail and ac quiring the ability to do this can be achieved o
138. e hypotheses contain no new variables Warning Save your work with save new_proof then write source before using depth 2 or greater since the search time grows exponentially and may never terminate in a reasonable time and you cannot interrupt the search I have found that it is rare for depth 3 or greater to be useful save new_proof Saves the proof in progress in the program s inter nal database buffer To save it permanently into the database file use write source after save new_proof To revert to the last save new_proof exit force from the Proof Assistant then re enter the Proof Assistant 5 6 CREATING PROOFS 141 match step step or match all Shows what statements are possi bilities for the assign statement This command is not very useful in its present form and hopefully will be improved eventually In the meantime use the search statement for candidates matching specific math token combinations minimize_with After a proof is complete this command will attempt to match other database theorems to the proof to see if the proof size can be reduced as a result See help minimize_with in the Metamath program for its usage The following commands set parameters that may be relevant to your proof Consult the individual help set commands set unification_timeout set search_limit set empty_substitution note that default is off Type exit to exit the MM PA gt prompt and get back to the MM gt prompt A
139. e simple enough so that virtually anyone should be able to un derstand them although that in itself will not let you play like a master The symbols that Metamath manipulates do not in themselves have any intrinsic meaning Your interpretation of the axioms that you supply to Metamath is what gives them meaning Metamath is an attempt to strip down mathematical thought to its bare essence and show you exactly how the symbols are manipulated Philosophers and logicians with various motivations have often thought it important to study weak fragments of logic 2 35 other unconven tional systems of logic such as modal logic 8 ch 27 and quantum logic in physics 44 Metamath provides a framework in which such sys tems can be expressed with an absolute precision that makes all underlying metamathematical assumptions rigorous and crystal clear Some schools of philosophical thought for example intuitionism and con structivism demand that the notions underlying any mathematical system be as simple and concrete as possible Metamath should meet the require ments of these philosophies Metamath must be taught the symbols axioms and rules for a specific theory from the skeptical such as intuitionis a to the bold such as the axiom of choice in set theory 27Intuitionism does not accept the law of excluded middle either something is true or it is not true See p xi for discussion and references on this topic
140. e you can lose a lot of work It is a good idea to periodically verify all proofs with verify proof to ensure their integrity If your file contains only normal as opposed to compressed proofs the main rule is that you may not change the order of the mandatory hypotheses of any statement referenced in a later proof For example if you swap the order of the major and minor premise in the modus ponens rule all proofs making use of that rule will become incorrect The show statement command will show you the mandatory hypotheses of a statement and their order If a statement has a compressed proof you also must not change the order of its mandatory hypotheses The compressed proof format makes use of this information as part of the compression technique Note that swapping the names of two variables in a theorem will change the order of its mandatory hypotheses The safest way to edit a statement say mytheorem is to duplicate it then rename the original to mytheoremOLD throughout the database Once the edited version is re proved all statements referencing mytheoremOLD can be updated in the Proof Assistant using minimize_with mytheorem allow_growth 54 CHAPTER 2 USING THE METAMATH PROGRAM Chapter 3 Abstract Mathematics Revealed 3 1 Logic and Set Theory Set theory can be viewed as a form of exact theology RUDY Rucked Despite its seeming complexity all of standard mathematics no matter how deep or abstract can amazi
141. ea of rigorously archiving mathematical knowledge in a computer database providing precision certainty and elimination of human error Note on Bibliography and Index The Bibliography usually includes the Library of Congress classification for a work to make it easier for you to find it in on a university library shelf Xiv PREFACE The Index has author references to pages where their works are cited even though the authors names may not appear on those pages Acknowledgments Acknowledgments are first due to my wife Deborah who passed away on September 4 1998 for critiquing the manuscript but most of all for her patience and support I also wish to thank Joe Wright Richard Becker Clarke Evans Buddha Buck and Jeremy Henty for helpful comments Any errors omissions and other shortcomings are of course my responsibility Note Added June 22 2005 The original unpublished version of this book was written in 1997 and distributed via the web The present edition has been updated to reflect the current Metamath program and databases as well as more current URLs for Internet sites Thanks to Josh Purinton One Hand Clapping Mel L O Cat and Roy F Longton for pointing out typographical and other errors I have also benefitted from numerous discussions with Raph Levien who has extended Metamath s philosophy of rigor to result in his Ghilbert proof language nttp ghiTbert ore Robert Bob Solovay communicated a new re
142. eally determined to get to the bottom of this The first thing the book does is explain how the axioms came about which was to resolve Russell s paradox In fact that seems to be the main purpose of their existence that they supposedly can be used to derive all of mathematics seems irrelevant and is not even mentioned Well you go on You hope the book will explain to you clearly step by step how to derive things from the axioms After all this is the starting point of mathematics like a book that explains the basics of a computer programming language But something is missing You find you can t even understand the first proof or do the first exercise Symbols such as J and V permeate the page without any mention of where they came from or how to manipulate them the author assumes you are totally familiar with them and doesn t even tell you what they mean By now you know that 4 means there exists and V means for all but shouldn t the rules for manipulating these symbols be part of the axioms You still have no idea how you could even describe the axioms to a computer Certainly there is something much different here from the technical lit erature you re used to reading A computer language manual almost always explains very clearly what all the symbols mean precisely what they do and the rules used for combining them and you work your way up from there After glancing at four or five other such books you come to the
143. ed logical equivalence Unlike most traditional developments we have chosen not to have a separate symbol such as Df to mean is defined as Instead we will use the biconditional connective for this purpose as it lets us to use logic to manipulate definitions directly Here we state the properties of the biconditional connective with a carefully crafted a statement which effectively uses the biconditional connective to define itself The symbol can be eliminated from a formula using theorem bii which is derived later Define the biconditional connective df bi a Fa ped o7 y gt b gt 7 4 gt 9 gt 7 4 e gt v gt y gt p gt pe 6The term connective is informally used to mean a symbol that is placed between two variables or adjacent to a variable whereas a mathematical constant usually indicates a symbol such as the number 0 that may replace a variable or metavariable From Metamath s point of view there is no distinction between a connective and a constant both are constants in the Metamath language 3 4 A HIERARCHY OF DEFINITIONS 69 df bi a ph lt gt ps gt ph gt ps gt ps gt ph gt ph gt ps gt ps gt ph gt ph lt gt ps This theorem relates the biconditional connective to primitive connectives and can be used to eliminate the symbol from any wff bii p F porv er pov gt 7 gt bi
144. ed proof of the famous Schr der Bernstein theorem in set theory was incorrect p 148 This theorem states that if there exists a 1 to 1 so from set A into set B and vice versa then sets A and B have a 1 to 1 correspondence Although it sounds simple and obvious the standard proof is quite long and complex e In the early 1900 s Hilbert published a purported proof of the con tinuum hypothesis which was eventually established as unprovable by Cohen in 1963 I6 p 166 The continuum hypothesis states that no infinity transfinite cardinal number exists whose size or cardi nality is between the size of the set of integers and the size of the set of real numbers This hypothesis originated with German mathe matician Georg Cantor in the late 1800 s and his inability to prove it is said to have contributed to mental illness that afflicted him in his later years e An incorrect proof of the four color theorem was published by Kempe in 1879 p 582 it stood for 11 years before its flaw was discov ered This theorem states that any map can be colored using only four colors so that no two adjacent countries have the same color In 1976 the theorem was finally proved by the famous computer assisted proof of Haken Appel and Koch 57 Or at least it seems that way 36 A set is any collection of objects A function or mapping is a rule that assigns to each element of one set called the function s domain an element from a
145. eel for the tools utility we show a simple session where we create a file n txt with 3 lines add strings before and after each line and display the lines on the screen You can experiment with the various commands to gain experience with the tools utility MM gt tools Entering the Text Tools utilities Type HELP for help EXIT to exit TOOLS gt number Output file lt n tmp gt n txt First number lt 1 gt Last number lt 10 gt 3 Increment lt 1 gt TOOLS gt add Input output file n txt String to add to beginning of each line lt gt This is line String to add to end of each line lt gt The file n txt has 3 lines 3 were changed First change is on line 1 This is line 1 TOOLS gt type n txt This is line 1 This is line 2 This is line 3 TOOLS gt exit Exiting the Text Tools Type EXIT again to exit Metamath MM gt 154 CHAPTER 5 THE METAMATH PROGRAM Appendix A Math Symbol Tokens for Set Theory This Appendix lists the tokens math symbols used for basic set theory development up to complex numbers in the set mm database in order of appearance Next to each token we show the mathematical symbol that corresponds to it The set mm file has an explanation of the meaning of each symbol New definitions are added to set mm over time Consult the latest version at http metamath org for the most recent additions These symbols are defined in the set mm database file inside of a special comm
146. efinition However to justify the definition as not being creative we need to prove that 126 CHAPTER 4 THE METAMATH LANGUAGE P Q gt R is in fact not a wff in the original language and this is more difficult than in the case where we simply introduce a new symbol What constitutes a definition versus an axiom is sometimes arbitrary in mathematical literature For example the connectives V OR A AND and equivalent to in propositional calculus are usually considered de fined symbols that can be used as abbreviations for expressions containing the primitive connectives and 7 This is the way we treat them in the standard logic and set theory database set mm However the first three con nectives can also be considered primitive and axiom systems have been devised that treat all of them as such For example I7 p 35 presents one with 15 axioms some of which in fact coincide with what we have chosen to call definitions in set mm In certain subsets of classical propositional calcu lus such as the intuitionist fragment it can be shown that one cannot make do with just and but must treat additional connectives as primitive in order for the system to make sense In set theory recursive definitions define a newly introduced symbol in terms of itself The justification of recursive definitions using several recur sion theorems is the usually one of the first sophisticated proofs a student
147. ehnke F Backmann K Fladt and W S ss editors Fundamentals of Mathematics volume I The MIT Press Cambridge Massachusetts 1974 QA37 2 B413 J L Bell and M Machover A Course in Mathematical Logic North Holland Amsterdam 1977 QA9 B3953 Andrea Blass The interaction between category theory and set theory In John Walter Gray editor Mathematical Applications of Category Theory Proceedings of the Special Session on Mathematical Applica tions Category Theory 89th Annual Meeting of the American Mathe matical Society held in Denver Colorado January 5 9 1983 pages 5 29 Providence Rhode Island 1983 American Mathematical Society QA169 A47 1983 W W Bledsoe and D W Loveland editors Automated Theorem Prov ing After 25 Years Proceedings of the Special Session on Automatic Theorem Proving 89th Annual Meeting of the American Mathemati cal Society held in Denver Colorado January 5 9 1983 Providence Rhode Island 1983 American Mathematical Society QA76 9 A96 S64 1983 George S Boolos and Richard C Jeffrey Computability and Log ic Cambridge University Press Cambridge third edition 1989 QA9 59 B66 1989 John Campbell Programmer s Progress White Star Software Box 51623 Palo Alto CA 94303 1991 181 182 10 11 12 13 14 15 16 17 18 19 20 wo am BIBLIOGRAPHY Shang Ching Chou Proving elementary geometry theorems using Wu s
148. elp give you a higher level un derstanding The set mm database provides references that will get you started From there you will be on your way towards a very deep rigorous understanding of abstract mathematics The Metamath program can help you peruse a Metamath database whether you are trying to figure out how a certain step follows in a proof or just have a general curiosity We will go through some examples of the commands using the set mm database provided with the Metamath soft ware These should help get you started See Chapter 5 for a more detailed description of the commands Note that we have included the full spelling of all commands to prevent ambiguity with future commands In practice you may type just the characters needed to specify each command keyword unambiguously often just one or two characters per keyword and you don t need to type them in upper case First run the Metamath program as described earlier You should see the MM gt prompt Read in the set mm file MM gt read set mm Reading source file set mm 73689 lines 3543983 characters were read from set mm The source has 21443 statements 417 are a and 5989 are p No errors were found However proofs were not checked Type VERIFY PROOF if you want to check them Let s check the database integrity This may take a minute or two to run if your computer is slow 84 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED MM gt verify proof O 10 20
149. eme of set theory Vz x y we may not sub stitute the same variable for both x and y On the other hand the theorem scheme x y gt y x does not require that x and y be distinct so we do not require a distinct variable restriction although having one would cause no harm other than making the scheme less general A mandatory variable type hypothesis is one whose variable exists in a logical hypothesis or the assertion A provable pre statement defined below may require non mandatory variable type hypotheses that effectively introduce dummy variables for use in its proof Any number of dummy variables might be required by a specific proof indeed it has been shown by H Andr ka that there is no finite upper bound to the number of dummy variables needed to prove an arbitrary theorem in first order logic with equality having a fixed number n gt 2 of individual variables See also the Comment on p For this reason we do not set a finite size bound on the collections D and T although in an actual application Metamath database these will of course be finite increased to whatever size is necessary as more proofs are added Relationship to Metamath A pre statement of a formal system corre sponds to an extended frame in a Metamath database Section 4 2 7 The collections D T and H correspond respectively to the d f and e state ment collections in an extended frame The expression A corresponds to the a or p statement
150. ement being proved and 2 a proof step tagged with Z is referenced before the step tagged with the Z Just as in a normal proof under development Section 4 4 5 any step or subproof that is not yet known may be represented with a single White space does not have to appear between the and the upper case letters or other s representing the remainder of the proof Appendix C Metamath s Formal System C 1 Introduction Perfection is when there is no longer anything more to take away ANTOINE DE SAINT EXUPERY This appendix describes the theory behind the Metamath language in an abstract way intended for mathematicians Specifically we construct two set theoretical objects a formal system roughly a set of syntax rules axioms and logical rules and its universe roughly the set of theorems derivable in the formal system The Metamath computer language provides us with a way to describe specific formal systems and with the aid of a proof provided by the user to verify that given theorems belong to their universes To understand this appendix you need a basic knowledge of informal set theory It should be sufficient to understand for example Ch 1 of Munkres Topology 41 or the introductory set theory chapter in many text books that introduce abstract mathematics Note that there are minor notational differences among authors e g Munkres uses C instead of our C for subset We use included in
151. ement on a computer Again the axiom system we use in the set mm database avoids the complexity of this condition by effectively embedding the complexity into the axioms themselves 3 2 4 Set Theory Traditional Zermelo Fraenkel set theory has 10 axioms which can be ex pressed in the language of predicate calculus In this section we will list only the names and brief English descriptions of these axioms since we will give you the precise formulas used by the Metamath set theory database set mm later on In the descriptions of the axioms we assume that x y z w and v rep resent sets These are the same as the variables in our predicate calculus system above except that now we informally think of the variables as rang ing over sets Note that the terms object set element collection and family are synonymous as are is an element of is a member of 62 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED is contained in and belongs to The different terms are used for con venience for example a collection of sets is less confusing than a set of sets A set x is said to be a subset of y if every element of x is also an element of y we also say x is included in y The axioms are very general and apply to almost any conceivable math ematical object and this level of abstraction can be overwhelming at first To gain an intuitive feel it can be helpful to dr
152. encounters when learning set theory and there is a significant amount of implicit metalogic behind a recursive definition even though the definition itself is typically simple to state It is however possible to substitute one kind of complexity for another We can eliminate the need for metalog ical justification by defining the operation directly with an explicit but complicated expression then deriving the recursive definition directly as a theorem using a recursion theorem in reverse We do this in set mm as follows In set mm our goal was to introduce almost all definitions in the form of two expressions connected by either or where the thing being defined does not appear on the right hand side Quine calls this form a genuine or direct definition 48 p 174 which makes the definitions very easy to eliminate and the metalogic needed to justify them as simple as possible We achieved this goal in almost all cases Sometimes this makes the defini tions more complex and less intuitive For example the traditional way to define addition of natural numbers is to define an operation called successor which means plus one and is denoted by suc then define addition recursively with the two definitions n 0 n and m sucn suc m n Although this definition seems simple and obvious the method to eliminate the definition is not obvious in the second part of the definition addition is defined in terms of it
153. ent which is indicated by the appearance of the two character string t at the beginning of the comment see Section 5 8 1 p 148 This special comment is called a t comment or typesetting comment The definitions in the t comment are referenced by the write tex command to produce ATFX output If you add a new token to the set theory database or your own database you should also update the t comment if you want to create a ATEX output file The t comment is not needed for normal operation of the Metamath program but is referenced only when you open a TX output file with the open tex command The t comment consists of a series of ATEX definitions with the following syntax latexdef token string as latez string The fields are separated by white space blanks carriage returns etc although white space is not needed before the terminator Each definition should start on a new linef For example latexdef _ as subseteq 1This restriction of the current version of Metamath 0 07 30 may be removed in a future version but you should do it anyway for readability 155 156 APPENDIX A MATH SYMBOL TOKENS FOR SET THEORY defines the token _ as the BTEX symbol C which means subset The token string and latex string are the character strings for the to ken and the TEX definition of the token respectively enclosed in either double or single quotation marks The string enclosed in quotation marks may
154. entential logic set set difference set intersection set theory 2 19 29 56 on set theory database set mm 27 29 31 33 asia af 76 83 126 1 48 150 155 176 set union Shoenfield Joseph R simple declaration simple infinite sequence simple metatheorem 171 singleton 194 Solovay Robert Solow Daniel source buffer 46 138 146 source file 96 a 97 ey 134 special characters 92 stack 40 95 Stark Harold M statement in a formal system stylized epsilon 57 60 subclass subset substitution oan proper sy 63 69 LOO 170 variable ea substitution map substitution theorem successor 74 79 Swart E R 21 symbol 96 in a formal system 162 syntax rules Szpiro George Takeuti Gaisi Tarski Alfred tautology temporary variable term 34 37 text editor theorem i ran eeann 47 55 E766 oe theorem scheme tilde 119 121 token 38 a TH m ey topology 3 transfinite recursion 80 transitive class transitive set tree style proof Ke oy INDEX trusting computers 21 truth a z turnstile hamm 1 P Boe type 99 105 typesetting comment 98 IT9 I21 149 union D 73 75 universal class V universal quantifier V restricted universe of a formal system Unix file names unordered pair unordered tri
155. equivalent to d d d d d d aM ee ge NNS NS N H A A HHH lt Two or more simple d statements specifying the same variable pair are internally combined into a single d statement Thus the set of three statements d x y d x y d yx is equivalent to d x y Similarly compound d statements after being internally broken up internally have their common variable pairs combined For example the set of statements dx y A dxyB is equivalent to d x y dx A dyA dx y dxB dyB 4 2 THE BASIC KEYWORDS 103 which is equivalent to d x y d x A d y AS dxB d yB S Metamath automatically verifies that all d restrictions are met whenever it verifies proofs d statements are never referenced directly in proofs this is why they do not have labels but Metamath is always aware of which ones must be satisfied i e are active and will notify you with an error message if any violation occurs To illustrate how Metamath detects a missing d statement we will look at the following example from the set mm database da xz d yz Theorem to add distinct quantifier to atomic formula axi7eq p l x y gt A z2x y This statement has the obvious requirement that z must be distinct from x in theorem axi7eq that states x y gt Yz x y well obvious if you re a logician for otherwise we could conclude z y gt Yx x y which is false when free v
156. er set theory as we do in this appendix and only the underlying set theory with its single binary predicate would be defined directly in the Metamath language 170 APPENDIX C METAMATH S FORMAL SYSTEM Comment As we noted earlier the specific variables denoted by the symbols x y z Vr C VR C SM in Example 2 are not the actual vari ables of ordinary predicate calculus but should be thought of as metavari ables ranging over them For example a distinct variable restriction would be meaningless for actual variables of ordinary predicate calculus since two different actual variables are by definition distinct And when we talk about an arbitrary representative a Vr a is a metavariable in our expository language that ranges over metavariables which are primitives of our for mal system each of which ranges over the actual individual variables of predicate calculus which are never mentioned in our formal system The constant called var above is called set in the set mm database file but it means the same thing I felt that var is a more meaningful name in the context of predicate calculus whose use is not limited to set theory For consistency we stick with the name var throughout this Appendix even after set theory is introduced C 3 3 Free Variables and Proper Substitution In the system of Example 2 there are no primitive notions of free vari able and proper substitution Tarski 61 shows that this
157. ere made to the source assertion Variable Substituted with x Zz y x ph x y The target hypothesis before substitution was pm2 61dd 2 e ph gt ch The following substitutions were made to the target hypothesis Variable Substituted with ph A ZZ2 xX ch x y gt A zx y The disjoint variable restrictions of ax 16 can be seen from the show statement command The line that begins Its mandatory disjoint variable pairs are lists any d variable pairs in brackets MM gt show statement ax 16 full Statement 3033 is located on line 9338 of the file set mm Axiom of Distinct Variables a ax 16 a l A x x y gt ph gt A x ph Its mandatory hypotheses in RPN order are wph f wff ph vx f set x vy f set y Its mandatory disjoint variable pairs are lt x y gt The statement and its hypotheses require the variables x y ph The variables it contains are x y ph Since Metamath will always detect when d statements are needed for a proof you don t have to worry too much about forgetting to put one in it can always be added if you see the error message above If you put in unnecessary d statements the worse that will happen is that your theorem might not be as general as it could be and this may limit its use later on On the other hand when you introduce axioms a statements you must be very careful to properly specify the necessary associated d state ments since Metamath has no
158. erifies that the result matches the expected theorem It makes logical inferences based only on rules of logic that are contained in a set of axioms or first principles that you provide to it as the starting point for proofs The complete specification of the Metamath language is only four pages long Section 4 1 p 92 Its simplicity may at first make you may wonder how it can do much of anything at all But in fact the kinds of symbol manipulations it performs are the ones that are implicitly done in all math ematical systems at the lowest level You can learn it relatively quickly and have complete confidence in any mathematical proof that it verifies On the other hand it is powerful and general enough so that virtually any mathe matical theory from the most basic to the deeply abstract can be described with it Although in principle Metamath can be used with any kind of mathe matics it is best suited for abstract or pure mathematics that is mostly concerned with theorems and their proofs as opposed to the kind of math ematics that deals with the practical manipulation of numbers Examples of branches of pure mathematics are logic set theory number theory 2Logic is the study of statements that are universally true regardless of the objects being described by the statements An example is the statement if P implies Q then either P is false or Q is true 3Set theory is the study of general purpose mathematical
159. ers 5 1 Invoking Metamath Unix Linux and Mac OS X have a command line interface called the bash shell In Mac OS X select the Terminal application from Applica tions Utilities To invoke Metamath from the bash shell prompt assuming that the Metamath program is in the current directory type bash metamath To invoke Metamath from a Windows DOS or Command Prompt as suming that the Metamath program is in the current directory or in a directory included in the Path system environment variable type C metamath gt metamath 129 130 CHAPTER 5 THE METAMATH PROGRAM To use command line arguments at invocation the command line argu ments should be a list of Metamath commands surrounded by quotes if they contain spaces In Windows the surrounding quotes must be double not single quotes For example to read the database file set mm verify all proofs and exit the program type under Unix bash metamath read set mm verify proof exit Note that in Unix any directory path with s must be surrounded by quotes so Metamath will not interpret the as a command qualifier So if set mm is in the tmp directory use for the above example bash metamath read tmp set mm verify proof exit For convenience if the command line has one argument and no spaces in the argument the command is implicitly assumed to be read In this one special case s are not interpreted as command qualifiers
160. es the output of any theorem that has been flagged with a question mark placed in or in place of the date comment field at the end of its proof for example 31 Oct 00 This lets you strip out proofs under development so that a clean version of the database can be generated for official release Note Currently hypotheses are not stripped only p statements Spurious date comment fields of the suppressed theorems may also remain Be careful to use a different name for the clean version so that your work in progress won t be destroyed 5 4 Showing Status and Statements 5 4 1 show settings Command Syntax show settings This command shows the state of various parameters 5 4 2 show memory Command Syntax show memory This command shows the available memory left It is not be meaningful on most modern operating systems which have virtual memory 5 4 3 show labels Command Syntax show labels label match al1 linear This command shows the labels of a and p statements that match label match A in label match matches zero or more characters For example abc def will match all labels containing abc and ending with def Optional command qualifier all Include matches for e and f statement labels linear Display only one label per line This can be useful for building scripts in conjunction with the utilities under the tools command 5 4 4 show statement Command Syntax show statement label
161. etail to write a parser and proof verifier The specification is terse and it is probably hard to learn the language directly from it but we include it here for those impatient people who prefer to see everything up front before looking at verbose expository material Later sections explain this material and provide examples We will repeat the definitions in those sections and you may skip the next section at first reading and proceed to Section 4 2 p 96 You may want to come back to it to clarify any fine points 4 1 Specification of the Metamath Language Sometimes one has to say difficult things but one ought to say them as simply as one knows how G H Harpy 4 1 1 Preliminaries A Metamath database is built up from a top level source file together with any source files that are brought in through file inclusion commands see below The only characters that are allowed to appear in a Metamath source file are the 94 printable characters on standard ASCII keyboards which are digits upper and lower case letters and the following 32 special characters t h7 amp OC _L zt CILI ee a LEN plus the following non printable white space characters space tab car riage return line feed and form feed We will use typewriter font to display the printable characters A Metamath database consists of a sequence of three kinds of tokens separated by white space which is any sequence of one or more white space characte
162. etermine a unique unification In this case Metamath will ask you to pick the correct one 4 4 Extensions to the Metamath Language 4 4 1 Comments in the Metamath Language The commenting feature allows you to annotate the contents of a database Just as with most computer languages comments are ignored for the purpose of interpreting the contents of the database Comments effectively act as additional white space between tokens when a database is parsed A comment may be placed between any two tokens in a source file Comments have the following syntax text Here textis a string possibly empty of any characters in Metamath s char acter set p 92 except that the character strings and may not appear in tert Thus nested comments are not permitted Metamath will com plain if you give it This is a nested comment To compensate for this behavior I often change all s to s in sections of Metamath code I wish to comment out Math Symbols and Labels Inside Comments Inside of comments a string of tokens enclosed in grave accents will be converted to standard mathematical symbols during TeX output typeset ting according to the information in the special t comment in the database as described in Appendix A The first causes the output processor to enter math mode and the second one exits it Two consecutive grave accents are treated as a single actual grave accent both inside
163. etween them White space must surround the parentheses and the labels The left parenthesis tells Metamath that a compressed proof follows A normal RPN proof consists of just a sequence of labels and a parenthesis is not a legal character in a label The sequence of upper case letters corresponds to a sequence of integers with the following mapping Each integer corresponds to a proof step as described later A 1 B 2 T 20 UA 21 UB 22 UT 40 VA 41 VB 42 YT 120 159 160 APPENDIX B COMPRESSED PROOFS UUA 121 YYT 620 UUUA 621 etc In other words A through T represent a least significant digit in base 20 and U through Y represent zero or more most significant digits in base 5 where the digits start counting at 1 instead of the usual 0 With this scheme we don t need white space between these numbers In the design of the compressed proof format only upper case letters as opposed to say all printable ASCII characters other than were chosen so as not to collide with most text editor searches at the expense of a typical 20 compression loss The base 5 base 20 grouping as opposed to say base 6 base 19 was chosen by experimentally determining the grouping that resulted in best typical compression The letter Z identifies tags a proof step that is identical to one that occurs later on in the proof it helps shorten the proof by not requiring that identical proof steps be proved over and over
164. ext Only With Line Breaks to get a plain text file On some computer systems Metamath does not have the capability to print its output directly instead you send its output to a file using the open commands described later The way you print this output file depends on your computer Some computers have a print command whereas with others you may have to read the file into an editor and print it from there 1 American Standard for Coded Information Interchange 2Tt is recommended that all lines in a Metamath source file be 79 characters or less in length for compatibility among different computer terminals When creating a source file on an editor such as Word select a monospaced font such as Courier or Monaco to make this easier to achieve Better yet just use a plain text editor such as Notepad 33 34 CHAPTER 2 USING THE METAMATH PROGRAM If you want to print your Metamath source files with typeset formulas containing standard mathematical symbols you will need the ATRX type setting program which is widely and freely available for most operating systems It runs natively on Unix and Linux and can be installed on Win dows as part of the free Cygwin package http cygwin com You can also produce HTMIp web pages The help html command in the Metamath program will assist you with this feature 2 2 Your First Formal System 2 2 1 From Nothing to Zero To give you a feel for what the Metamath language looks like we wi
165. f of of of of of length length length length length length length length length and above CONGRATULATIONS NEW_PROOF to save OwWwroaorWrRRrR have been renumbered wff term was was was was was was was was was t at t a ct ct found found found found found found found found found for for for for for for for for for step step step step step step step step step gt t 0 t gt t t gt t t Ti 10 9 7 6 5 3 2 i The proof is complete Use SAVE The Proof Assistant does it N ote 52 CHAPTER 2 USING THE METAMATH PROGRAM not detect d violations After saving the proof you should verify it with VERIFY PROOF The save new_proof command will save the proof in the database Here we will just display it in a form that can be clipped out of a log file and inserted manually into the database source file with a text editor MM PA gt show new_proof normal Secsacse Clip out the proof below this line tt tze tpl tt weq tt tt weq tt a2 tt tze tpl tt weq tt tze tpl tt weq tt tt weq wim tt a2 tt tze tpl tt tt ai mp mp SSSSSSsS The proof of thi to clip out ends above this line There is another proof format called compressed that you will see in databases It is not important to understand how it is encoded but only to recognize it when you see it Its only purpose is to reduce storage requirements for large proofs A com
166. f metavariable and which is provable by combining cases outside the system as above A system is metalogically complete if all of its simple metatheorems are directly provable statements The precise definition of simple metatheorem and the proof of the metalogical completeness of Example 3 is found in Remark 9 6 and Theorem 9 7 of 34 C 3 5 Example 3 Metalogically Complete Predicate Calculus with Equality For simplicity we will assume there is one binary predicate R this system suffices for set theory where the R is of course the predicate We label the axioms as they appear in 34 This system is logically equivalent to that of Example 2 when the latter is restricted to this single binary predicate but is also metalogically complete Let CN wff var gt 7 V R VR 9 Xx U z y 2 T wff y wE Y wE x U var z var y var 2 Then T consists of the reducts of the following pre statements D T 2 wif p gt 172 APPENDIX C METAMATH S FORMAL SYSTEM wif T wif Vax p D T D T wE Ray y T Cl 2 T 2 F p gt Y gt C2 2 T E p gt YW gt x gt YY gt 8 gt x C3 2 T F Cy gt Y gt V gt C4 2 T F Vary gt Y gt Yr y gt Vz Y C5 8 T F Yzy gt p C6 T F VaVy y g
167. f the file demoO mm Define the modus ponens inference rule 17 mp a Q Its mandatory hypotheses in RPN order are wp f wff P wq f wif Q min e P maj e P gt Q The statement and its hypotheses require the variables QP The variables it contains are Q P The mandatory hypotheses and their order are useful to know when you are trying to understand or debug a proof Now you are ready to look at what s really inside of our proof First here is how to look at every step in the proof not just the ones corresponding to an ordinary formal proof but also the ones that build up the formulas that appear in each ordinary formal proof step 2 3 A TRIAL RUN 43 MM gt show proof thi lemmon all This will display the proof on the screen in the following format 1 tt f term t 2 tze a term 0 3 1 2 tpl a term t 0 4 tt f term t 5 3 4 weq a wff t 0 t 6 tt f term t 7 tt f term t 8 6 7 weq a wff t t 9 tt f term t 10 9 a2 a l t 0 t 11 tt f term t 12 tze a term 0 13 11 12 tpl a term t 0 14 tt f term t 15 13 14 weq a wff t 0 t 16 tt f term t 17 tze a term 0 18 16 17 tpl a term t 0 19 tt f term t 20 18 19 weq a wff t 0 t 21 tt f term t 22 tt f term t 23 21 22 weq a wff t t 24 20 23 wim a wff t 0 t gt t t 25 tt f term t 26 25 a2 a l t 0 t 27 tt f term t 28 tze a term 0 29 27 28 tpl a term t 0 30 tt f term t 31 tt f term t 32 29 3
168. f them as otherwise unspecified sets that happen to satisfy the axioms For the axioms we are given or postulate 8 classes C the set of complex numbers R the set of real numbers a subset of C 0 zero 1 one i square root of 1 plus times and lt less than Subtraction and division are defined terms and are not part of the axioms for their definitions see set mm 3 7 AXIOMS FOR REAL AND COMPLEX NUMBERS 81 Note that the notation A B and similarly A B specifies a class called an operation and is the function value of the class at ordered pair A B An operation is defined by statement df opr on p 76 The notation A lt B specifies a wff called a binary relation and means A B lt as defined by statement df br on p Our set of 8 given classes is assumed to satisfy the following 28 axioms pat The class of complex numbers is a set axcnex p FCeEV 2 The real numbers are a subset of the complex numbers axresscn p RCC 3 Zero is a real number axOre p FOER 4 One is a real number axire p F1ER 5 The imaginary number 7 is a complex number axicn p FicC 6 Complex numbers are closed under addition axaddcl p AGCCABEC gt A B EC 7 Real numbers are closed under addition axaddrcl p H AERABER gt A B ER 8 Complex numbers are closed under multiplication axmulcl p H AECABEC gt A B ecC 9 Real numbers are closed under multiplication axmulrcl p H AERABER gt
169. file discarding lines occurring exactly once unique Extract lines occurring exactly once in a file type 10 lines Display the first few lines in a file Similar to Unix head copy Similar to Unix cat but safe same input and output file allowed submit Run a script containing tools commands Note unduplicate duplicate and unique also sort the lines as a side effect 5 9 3 Using tools to Build Metamath submit Scripts The break command is typically used to break up a series of statement labels such as the output of Metamath s show usage into one label per line The other tools commands can then be used to add strings before and after each statement label to specify commands to be performed on the statement The parallel command is useful when a statement label must be mentioned more than once on a line 5 9 TEXT FILE UTILITIES 153 Very often a submit script for Metamath will require multiple command lines for each statement being processed For example you may want to enter the Proof Assistant minimize_with your latest theorem save the new proof and exit the Proof Assistant To accomplish this you can build a file with these four commands for each statement on a single line separating each command with a designated character such as Then at the end you can substitute each with n to break up the lines into individual command lines see help substitute 5 9 4 Example of a tools Session To give you a quick f
170. four color theorem a map needs no more than four colors to prevent any two adjacent countries from having the same color can presently only be done with the aid of a very complex computer program which originally required 1200 hours of computer time There has been considerable debate about whether such a proof can be trusted and whether such a proof is real mathematics 57 However under normal circumstances even a skeptical mathematician would a have a great deal of confidence in the result of multiplying two numbers on a pocket calculator even though the precise details of what goes on are hidden from its user Even the verification on a supercomputer 32 As quoted in i p 121 33 62 p 245 22 CHAPTER 1 INTRODUCTION that a huge number is prime is trusted especially if there is independent verification no one bothers to debate the philosophical significance of its proof even though the actual proof would be so large that it would be completely impractical to ever write it down on paper It seems that if the algorithm used by the computer is simple enough to be readily understood then the computer can be trusted Metamath adopts this philosophy The simplicity of its language makes it easy to learn and because of its simplicity one can have essentially absolute confidence that a proof is correct All axioms rules and definitions are available for inspection at any time because they are defined by the user
171. ge as far back as he or she wants all the way to the axioms on which the theorem is based This ability suggests a non traditional way of learning about pure mathematics Used in conjunction with traditional methods Metamath could make pure mathematics accessible to people who are not sufficiently skilled to figure out the implicit detail in ordinary text book proofs Once you learn the axioms of a theory you can have complete confidence that everything you need to understand a proof you are studying is all there at your beck and call allowing you to focus in on any proof step you don t understand in as much depth as you need without worrying about getting stuck on a step you can t figure out f 5Group theory studies the properties of mathematical objects called groups that obey a simple set of axioms and have properties of symmetry that make them useful in many other fields 6 Abstract algebra includes group theory and also studies groups with additional prop erties that qualify them as rings and fields The set of real numbers is a familiar example of a field T Analysis is the study of real and complex numbers 8One area studied by topology are properties that remain unchanged when geometrical objects undergo stretching deformations for example a doughnut and a coffee cup each have one hole the cup s hole is in its handle and are thus considered topologically equivalent In general though topology is the study of a
172. ght from Ancient to Modern Times Oxford University Press New York 1972 QA21 K516 1990 v 3 Morris Kline Mathematics The Loss of Certainty Oxford University Press New York 1980 QA21 K525 Edna E Kramer The Nature and Growth of Modern Mathematics Princeton University Press Princeton New Jersey 1981 QA93 K89 1981 Edmund Landau Foundations of Analysis Chelsea Publishing Com pany New York second edition 1960 QA241 L2541 1960 Hugues Leblanc On Meyer and Lambert s quantificational calculus FQ The Journal of Symbolic Logic 33 275 280 1968 QA J87 Czeslaw Lejewski On implicational definitions Studia Logica 8 189 208 1958 B18 P6 S933 Adrian R D Mathias A term of length 4 523 659 424 929 Synthese 133 75 86 2002 Q S993 Norman D Megill A finitely axiomatized formalization of predicate calculus with equality Notre Dame Journal of Formal Logic 36 435 453 1995 QA N914 Norman D Megill and Martin W Bunder Weaker D complete logics Journal of the IGPL 4 215 225 1996 Available on the Web at http wiw mpi sb mpg de igpl Journal V4 2 Megill Elliott Mendelson Introduction to Mathematical Logic D Van Nos trand Company Inc New York second edition 1979 QA9 M537 1979 184 37 38 39 40 41 42 43 44 45 46 47 48 49 50 BIBLIOGRAPHY C A Meredith Single axioms for the systems C N C O and A N of the tw
173. h this feature enabled more unifications will be ambiguous in the middle of a proof because substitution of variables with empty symbol se quences will become an additional possibility Let s see what happens when we make our next assignment MM PA gt assign 3 a2 There are 2 possible unifications Please select the correct one or Q if you want to UNIFY later Unify 6 with 9 0 9 Unification 1 of 2 weight 7 Replace 6 with 0 Replace 9 with Accept A reject R or quit Q lt A gt r The first choice presented is the wrong one If we had selected it tempo rary variable 6 would have been assigned a truncated wff and temporary variable 9 would have been assigned an empty sequence which is not al lowed in our system With this choice eventually we would reach a point where we would get stuck because we would end up with steps impossible to prove You may want to try it We typed r to reject the choice Unification 2 of 2 weight 21 Replace 6 with 9 0 9 Accept A reject R or quit Q lt A gt q To undo the assignment DELETE STEP 4 and INITIALIZE UNIFY if needed 7 min 8 8 maj 8 gt 6 gt t t The second choice is correct and normally we would type a to accept it But instead we typed q to show what will happen it will leave the step with an unknown unification which can be seen as follows 50 CHAPTER 2 USING THE METAMATH PR
174. hesis is mp a step 34 The source assertion before substitution was weq a wff t r The following substitutions were made to the source assertion Variable Substituted with t t 0 r t The target hypothesis before substitution was wp f wff P The following substitution was made to the target hypothesis Variable Substituted with P t 0 t The full proof just shown is useful to understand what is going on in de tail However most of the time you will just be interested in the essential or logical steps of a proof i e those steps that correspond to an ordinary formal proof If you type MM gt show proof thi essential lemmon renumber you will see 1 a2 a l C t 0 t 2 a2 a l t 0 t 3 al a l CC t 0 t gt CC t 0 t gt t t 4 2 3 mp a l CC t 0 t gt t t 5 1 4 mp a l t ll ct 2 3 A TRIAL RUN 45 Compare this to the formal proof on p 35 and notice the resemblance The essential qualifier in the show proof command tells Metamath to dis card all f hypotheses and everything branching off of them in the proof tree when the proof is displayed this makes the proof look more like an ordinary mathematical proof which does not normally incorporate the explicit con struction of expressions The renumber qualifier means to renumber the steps to correspond only to what is displayed To exit Metamath type MM gt exit 2 3 1 Some Hints for Using the Command Line Inter fac
175. hich by now you ve resigned yourself to treating as rather complex additional axioms since they can t be proved from the axioms you were given You can now get the computer to verify and even generate complete rigorous formal proofs Never mind that they may have 100 000 steps at least now you can have complete absolute confidence in them Unfortunately the only theorems you have proved are pretty trivial and you can easily verify them in a few minutes with truth tables if not by inspection It looks like your mathematician friend was right Getting the computer to do serious mathematics with this kind of rigor seems almost hopeless Even worse it seems that the further along you get the more axioms you have to add as each new theorem seems to involve additional meta mathematical reasoning that hasn t been formalized and none of it can be derived from the axioms of logic Not only do the proofs keep growing exponentially as you get further along but the program to verify them keeps getting bigger and bigger as you program in more metatheorems 4 The bugs that have cropped up so far have already made you start to lose faith in the rigor you seem to have achieved and you know it s just going to get worse as your program gets larger KKK 1 1 2 Mathematics and the Non Specialist A real proof is not checkable by a machine or even by any math ematician not privy to the gestalt the mode of thought of the part
176. i p l ph lt gt ps lt gt ph gt ps gt ps gt ph Define disjunction OR df or a H pvy e 7y gt y df or a ph V ps lt gt ph gt ps Define conjunction AND df an a H p y gt 7 y gt y df an a ph ps lt gt ph gt ps Define disjunction OR of 3 wffs df 30r a F yvVyvx e yvy vx df 30r a ph V ps V ch lt gt ph ps V ch Define conjunction AND of 3 wffs df 3an a F pApax e pay ax df 3an a ph ps ch lt gt ph ps ch 3 4 2 Definitions for Predicate Calculus The symbols z y and z represent individual variables of predicate calcu lus In this section they are not necessarily distinct unless it is explicitly mentioned Define existential quantification df ex a F daporVaeny df ex a E x ph lt gt A x ph Define proper substitution Note that this definition is valid even when x and y are the same variable The first conjunct is a trick used to achieve this property making the definition look somewhat peculiar at first For our notation we use y x p to mean the wff that results when y is properly substituted with x in the wff y The notation is different from the notation p x y that is sometimes used because the latter notation is ambiguous for us for example we don t know whether y aly is to be inter
177. i e it is the only function that satisfies tfr1 and tfr2 Note that f is an individual variable like x and y it is just a mnemonic to remind us that A is a collection of functions tfr 1 e KA f dxeOn fFnxAVyexn fiy G S fly tfir 2 e FF UA tfr1 p FFnOn tfr2 p ze On gt F z G Ffz tfr3 p BFnOnAV eeOn Bix G Blax gt B F The existence of omega the class of natural numbers Axiom 7 of Takeuti and Zaring p 43 This is the only theorem in this section requiring the Axiom of Infinity omex p FwEeV 3 7 Axioms for Real and Complex Numbers This section presents the axioms for real and complex numbers Analysis textbooks implicitly or explicity use these axioms or their equivalents are used as their starting point In the database set mm we define real and complex numbers as rather complicated specific sets and derive the these axioms as theorems from the axioms of ZF set theory using a method called Dedekind cuts We omit the details of this construction which you can fol low if you wish using the set mm database in conjunction with the textbooks referenced therein The construction is actually unimportant other than to show that sets exist that satisfy the axioms and thus that the axioms are consistent if ZF set theory is consistent When working with real numbers you can think of them as being the actual sets resulting from the construc tion for definiteness or you can think o
178. iable such as 1 in a proof It is useful for helping difficult unifications and it is necessary when you have dummy variables that eventually must be assigned a name let step step symbol sequence Forces a symbol sequence to replace the contents of a proof step provided it can be unified with the existing step contents I rarely use this unify step step or unify all Unifies the source and target of a step If you specify a specific step you will be prompted to select among the unifications that are possible If you specify all all steps with unique unifications but only those steps will be unified unify all interactive goes through all non unified steps initialize step or all De unifies the target and source of a step or all steps as well as the hypotheses of the source and makes all variables in the source unknown Useful to recover from an assign or let mistake that resulted in incorrect unifications delete step or all or floating_hypotheses Deletes the specified step s delete floating_hypotheses then initialize all then unify all interactive is useful for recovering from mistakes where incorrect unifications assigned wrong math symbol strings to variables improve step or all Automatically creates a proof for steps with no unknown variables whose proof requires no statements with e hypotheses Useful for filling in proofs of f hypotheses The depth qualifier will also try statements whose
179. iawez 2AVw wez7 rVvu VvE z gt nvew AAVeVw A zerrrnwesr o 47zE w gt 4Az uU gt Ez J J ArAVy ACA Vzez AVw zeEwrr wey zEy gt V2z Vw wezFowey OVwrwez F7AVw wezr74Vu vez770 w A7Vz2zVw gt a ze yrrwey o 7z2 wo 47z w gt wez Joa Vzrz yrorVw a weyorV2z wez745z y A7 AVz wez7 gt 71zey rwey 7zEy Section 3 4 shows the hierarchy of definitions that leads up to this expression 58 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED set theory intuition can be misleading for infinite collections and rigorous proofs become more important For example while y y x is correct for finite ordinals which are the natural numbers it is not usually true for infinite ordinals 3 2 The Axioms for All of Mathematics In this section we will show you how the axioms for all of standard math ematics i e logic and set theory as they are traditionally presented The traditional presentation is useful for someone with the mathematical ex perience needed to correctly manipulate high level abstract concepts For someone without this talent knowing how to actually make use of these ax ioms can be difficult The purpose of this section is to allow you to see how the version of the axioms used in the standard Metamath database set mm relates to the typical version in textbooks and also to give you an informal feel for them 3 2 1 Propositional Calculus Propositional calculus concer
180. ibly empty The basic Metamath language has two kinds of math symbols con stants and variables In a Metamath proof a constant may not be sub stituted with any expression A variable can be substituted with any ex pression This sequence may include other variables and may even include the variable being substituted This substitution takes place when proofs are verified and it will be described in Section 4 3 The f statement de scribed later in Section 4 2 5 is used to specify the type of a variable i e what kind of variable it is and give it a meaning typically associated with a metavariable in ordinary mathematics for example a variable may be specified to be a wff or well formed formula in logic a set in set theory or a non negative integer in number theory 4 2 3 The c and v Declaration Statements Constants are introduced or declared with c statements and variables are declared with v statements A simple declaration statement introduces a single constant or variable Its syntax is one of the following c math symbol v math symbol The notation math symbol means any math symbol token Some examples of simple declaration statements are c c gt c v x v y2 The characters in a math symbol being declared are irrelevant to Meta math for example we could declare a right parenthesis to be a variable 8A metavariable is a variable that ranges over the syntact
181. ical elements of the object language being discussed for example one metavariable might represent a variable of the object language and another metavariable might represent a formula in the object language 100 CHAPTER 4 THE METAMATH LANGUAGE v although this would be unconventional A compound declaration statement is a shorthand for declaring several symbols at once Its syntax is one of the following c math symbol math symbol v math symbol math symbol Here the ellipsis means any number of math symbols An example of a compound declaration statement is v x y mu This is equivalent to the three simple declaration statements v x v y v mu There are certain rules on where in the database math symbols may be declared what sections of the database are aware of them i e where they are active and when they may be declared more than once These will be discussed in Section 4 2 8 and specifically on p 113 4 2 4 The d Statement The d statement is called a disjoint variable restriction The syntax of the simple version of this statement is d variable variable where each variable is a previously declared variable and the two variables are different More specifically each variable must be an active variable which means there must be a previous v statement whose scope includes the d statement These terms will be defined when we discuss scoping statements in
182. ications 67 This program not only is able to generate relatively efficient proofs it can even be instructed to search for shorter proofs The effective use of OTTER does require a certain amount of experience skill and patience The axiom system used in the set mm set theory database can be expressed to OTTER using a method described in When successful this method tends to generate short and clever proofs but my experiments with it indicate that the method will find proofs within a reasonable time only for relatively easy theorems It is still fun to experiment with Reference 7 surveys a number of approaches people have explored in the field of automated theorem proving 1 3 3 Proof Verifiers A proof verifier is a program that doesn t generate proofs but instead verifies proofs that you give it Many proof verifiers have limited built in automated proof capabilities such as figuring out simple logical inferences while still being guided by a person who provides the overall proof Metamath has no built in automated proof capability other than the limited capability of its Proof Assistant Proof verification languages are not used as frequently as they might be Pure mathematicians are more concerned with producing new results and such detail and rigor would interfere with that goal The use of computers in http www cl cam ac us Ressar ch NUG Isabelle http j F http www cs unm edu nccune otter o use those axi
183. icular field of mathematics in which the proof is located DAVIS AND Hersu 25 The bulk of abstract or theoretical mathematics is ordinarily outside the reach of anyone but a few specialists in each field who have completed the necessary difficult internship in order to enter its coterie The typical intelligent layperson has no reasonable hope of understanding much of it nor even the specialist mathematician of understanding other fields It is like a foreign language that has no dictionary to look up the translation the only way you can learn it is by living in the country for a few years It is argued that the effort involved in learning a specialty is a necessary process 14A metatheorem is usually a statement that is too general to be directly provable in a theory For example if n1 n2 and ng are integers then n n2 ng is an integer is a theorem of number theory But for any integer k gt 1 if n1 n are integers then n n is an integer is a metatheorem in other words a family of theorems one for every k The reason it is not a theorem is that the general sum n nx asa function of k is not an operation that can be defined directly in number theory 15 T3 p 354 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 13 for acquiring a deep understanding Of course this is almost certainly true if one is to make significant contributions to a field in particular doing proofs is probably the most
184. in mathematics is considered to be a set A set called the empty set contains nothing We represent the empty set by Many sets can be built up from the empty set There is a set represented by that contains the empty set another set represented by that contains this set as well as the empty set another set represented by that contains just the set that contains the empty set and so on ad infinitum All mathematical objects no matter how complex are defined as being identical to certain sets the integer 0 is defined as the empty set the integer 1 is defined as the integer 2 is defined as 9 How these definitions were chosen doesn t matter now but the idea behind it is that these sets have the properties we expect of integers once suitable operations are defined Mathematical operations such as addition are defined in terms of operations on sets their union intersection and so on operations you may have used in elementary school when you worked with groups of apples and oranges With a leap of faith the axioms also postulate the existence of infinite sets such as the set of all non negative integers 0 1 2 also called natu ral numbers This set can t be represented with the brace notation we just showed you but requires a more complicated notation called class abstrac tion For example the infinite set x x is a natural number means the set of all objects x such that
185. in ordinary mathe matics partly because the variables of Metamath are not really the same as the variables of ordinary mathematics but rather are metavariables ranging over them as well as over other kinds of symbols and groups of symbols Depending on the situation we may informally interpret the d statement in different ways Suppose for example that x and y are variables ranging over numbers more precisely that x and y are metavariables ranging over variables that range over numbers and that ph p and ps w are variables more precisely metavariables ranging over formulas We can make the fol lowing interpretations that correspond to the informal language of ordinary mathematics 102 CHAPTER 4 THE METAMATH LANGUAGE d x y means assume x and y are distinct variables d x ph means assume x does not occur in y d ph ps means assume y and have no variables in common Compound d Statements The compound version of the d statement is a shorthand for specifying several variables whose substitutions must be pairwise disjoint Its syntax is d variable variable Here variable represents the token of a previously declared variable specif ically an active variable and all variables are different The compound d statements is internally broken up by Metamath into one simple d state ment for each possible pair of variables in the original d statement For example dwxyaz is
186. in the case of d or an expression for the other two keywords The math symbol string following a a source statement cor responds to expression A in an axiomatic statement of the formal system the one following a p source statement corresponds to A in a provable statement that is not axiomatic In other words each extended frame in a Metamath database corresponds to a pre statement of the formal system and a frame corresponds to a statement of the formal system Don t confuse the two meanings of statement here A statement of the formal system corresponds to the several statements in a Metamath database that may constitute a frame In order for the computer to verify that a formal system statement is provable each p source statement is accompanied by a proof However the proof does not correspond to anything in the formal system but is simply a way of communicating to the computer the information needed for its verification The proof tells the computer how to construct specific members of closure of the formal system pre statement corresponding to the extended frame of the p statement The final result of the construction is the member of the closure that matches the p statement The abstract formal system on the other hand is concerned only with the existence of members of the closure As mentioned on p Examples 1 and 3 6 in the previous Section parallel the development of logic and set theory in the Metamath database set m
187. inference are allowed Metamath is not limited to any specific field of mathematics The Metamath language is simple and robust with an almost total absence of hard wired syntax and I believe that it provides about the simplest possible framework that allows essentially all of mathematics to be expressed with absolute rigor Using the Metamath language you can build formal or mathematical systemd that involve inferences from axioms Although a database is pro vided that includes a recommended set of axioms for standard mathematics if you wish you can supply your own symbols syntax axioms rules and definitions The name Metamath was chosen to suggest that the language provides a means for describing mathematics rather than being the mathematics itself Actually in some sense any mathematical language is metamathematical Symbols written on paper or stored in a computer are not mathematics itself but rather a way of expressing mathematics For example 7 and VII are symbols for denoting the number seven in Arabic and Roman numerals neither is the number seven If you are able to understand and write computer programs you should be able to follow abstract mathematics with the aid of Metamath Used in conjunction with standard textbooks Metamath can guide you step by step towards an understanding of abstract mathematics from a very rigorous 1A formal or mathematical system consists of a collection of symbols such as
188. ing a proof when you know what should be in the proof step but the unification algorithm doesn t yet have enough information to completely specify the temporary variables A temporary variable is one that has the form nn in the proof display such as 1 2 etc The symbol sequence may contain other unknown variables if desired Examples let variable 32 A B let variable 32 A 35 let step 10 x x let step 2 7 ph Any symbol sequence will be accepted for the let variable command Only those symbol sequences that can be unified with the step will be ac cepted for let step The let commands zap the proof with information that can only be verified when the proof is built up further If you make an error the command sequence delete floating_hypotheses initialize all and unify all interactive will undo a bad let assignment If step is zero or negative the stepth from last unknown step as shown by show new_proof unknown will be used The command let step 0 symbol sequence will use the last unknown step let step 1 symbol sequence the penultimate etc If step is positive let step may be used to assign known in the sense of having previously been assigned a label with assign as well as unknown steps Either single or double quotes can surround the symbol sequence as long as they are different from any quotes inside of symbol sequence If symbol sequence contains both kinds
189. ion 2 b describes how distinct variable restrictions in the ax iomatic statement must be met It means that after a substitution for two variables that must be distinct the resulting two expressions must either contain no variables or if they do they may not have variables in common and each pair of any variables they do have with one variable from each expression must be specified as distinct in the original statement Relationship to Metamath Axiomatic statements and provable state ments in a formal system correspond to the frames for a and p statements respectively in a Metamath database The set of axiomatic statements are a subset of the set of provable statements in a formal system although in a Metamath database a a statement is distinguished by not having a proof A Metamath language proof for a p statement tells the computer how to explicitly construct a series of members of the closure ultimately leading to a demonstration that the assertion being proved is in the closure The actual closure typically contains an infinite number of expressions A formal system itself does not have an explicit object called a proof but rather the existence of a proof is implied indirectly by membership of an assertion in a provable statement s closure We do this to make the formal system easier to describe in the language of set theory 3This definition of closure incorporates a simplification due to Josh Purinton 166 APPENDIX C
190. ion will give you only a vague notion of what the keywords are for later we will describe the keywords in detail Table 4 1 Summary of the basic Metamath keywords Keyword Description c Constant symbol declaration v Variable symbol declaration d Disjoint variable restriction f Variable type floating hypothesis e Logical essential hypothesis a Axiomatic assertion p Provable assertion Start of proof in p statement End of the above statement types Start of block End of block There are some additional keywords called auxiliary keywords that 4 2 THE BASIC KEYWORDS 97 help make Metamath more practical These are part of the extended lan guage They provide you with a means to put comments into a Metamath source file and reference other source files We will introduce these in later sections Table 4 2 summarizes them so that you can recognize them now if you want to peruse some source files while learning the basic keywords Table 4 2 Auxiliary Metamath keywords Keyword Description Start of comment End of comment Start of included source file name End of included source file name Unlike those in some computer languages the keywords are short two character sequences rather than English like words While this may make them slightly more difficult to remember at first their brevity allows them to ble
191. ions are computed which when made to the referenced assertion s mandatory hy potheses cause these hypotheses to match the stack entries These same substitutions are then made to the variables in the referenced assertion it self which is then pushed onto the stack At the end of the proof there should be one stack entry namely the assertion being proved This process is explained in detail in Section 4 3 Metamath s proof notation is not very readable for humans but it allows the proof to be stored compactly in a file The Metamath program has proof display features that let you see what s going on in a more readable way as you will see in the next section The rules used in verifying a proof are not based on any built in syntax of the symbol sequence in an assertion nor on any built in meanings attached to specific symbol names They are based strictly on symbol matching con stants must match themselves and variables may be replaced with anything that allows a match to occur For example instead of term 0 and we could have just as well used yellow zero and provable as long as we did so consistently throughout the database Also we could have used is provable two tokens instead of one token throughout the database In each of these cases the proof would be exactly the same The inde pendence of proofs and notation means that you have a lot of flexibility to change the notation you use without having to change any pr
192. is a function from integers the function s domain to their squares its range If we restrict the domain to even integers its range will become the set of squares of even integers so this axioms asserts that the set of squares of even numbers exists Technical note In general the function need not be a set but can be a proper class 10 Axiom of Choice Let x be a set whose members are pairwise disjoint i e whose members contain no elements in common Then there exists another set containing one element from each member of x For example if x is y z w v where y z w and v are different sets then a set such as z w exists but the axiom doesn t tell us which one Actually the Axiom of Choice is redundant if the set x as in this example has a finite number of elements 3 3 THE AXIOMS IN THE METAMATH LANGUAGE 63 The Axiom of Choice is usually considered an extension of ZF set theory rather than a proper part of it It is sometimes considered philosophically controversial because it specifies the existence of a set without specifying what the set is ZF set theory that includes the Axiom of Choice is called ZFC When expressed symbolically the Axiom of Separation and the Axiom of Replacement contain wff symbols and therefore each represent infinitely many axioms one for each possible wff For this reason they are often called axiom schemes It turns out that the Axiom of the Null Set the Axiom of Pairing an
193. is to be done In principle any Metamath database can be converted to the above format and the above format is the most convenient to use when studying a Metamath database as a formal system Appendix C In fact Metamath internally converts the database to the above format The command show statement in the Metamath program will show you 112 CHAPTER 4 THE METAMATH LANGUAGE the contents of the frame for any a or p statement as well as its extension in the case of a p statement During our discussion of scoping statements it may be helpful to think in terms of the equivalent sequence of frames that will result when the database is parsed Scoping other than the limited use above to delimit frames is not a theoretical requirement for Metamath but makes it more convenient 4 2 8 Scoping Statements and The scoping statements start of block and end of block provide a means for controlling the portion of a database over which certain statement types are recognized The syntax of a scoping statement is very simple it just consists of the statement s keyword For example consider the following database where we have stripped out all tokens except the scoping statement keywords For the purpose of the discussion we have added subscripts to the scoping statements these subscripts do not appear in the actual database 1 2 ho 3 4 4 s H Each statement in this example is said to be matched wi
194. it s hard to pin down precisely what the rules are Sure 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 7 each step seems to follow logically from the others but exactly what does that mean Is the logic just a matter of common sense something vague that we all understand but can never quite state precisely You ve spent a number of years off and on programming computers and you know that in the case of computer languages there is no question of what the rules are they are precise and crystal clear If you follow them your program will work and if you don t it won t No matter how complex a program it can always be broken down into simpler and simpler pieces until you can ultimately identify the bits that are moved around to perform a specific function Some programs might require a lot of perseverance to accomplish this but if you focus on a specific portion of it you don t even necessarily have to know how the rest of it works Shouldn t there be an analogy in mathematics You decide to apply the ultimate test you ask yourself how a computer could verify or ensure that the steps in these proofs follow from one another Certainly mathematics must be at least as precisely defined as a computer language if not more so after all computer science itself is based on it If you can get a computer to verify these proofs then you should also be able in principle to understand them yourself in a very crystal clear precise wa
195. iting for your input You will see the following on your screen Metamath Version 0 07 30 8 Feb 2007 Type HELP for help EXIT to exit MM gt The MM gt prompt means that Metamath is waiting for a command The help message line suggests that commands should be typed in upper case but actually command keywords are not case sensitive We will use lower case in our examples The first thing that you need to do is to read in your database MM gt read demoO mm Remember to press the return key after entering this command If you omit the file name Metamath will prompt you for one The syntax for specifying a Macintosh file name path is given in a footnote on p 123 If there are any syntax errors in the database Metamath will let you know when it reads in the file The one thing that Metamath does not check when reading in a database is that all proofs are correct because this would slow it down too much It is a good idea to periodically verify the proofs in a database you are making changes to To do this use the following command and do it for your demo0 mm file now Note that the is a wild card meaning all proofs in the file MM gt verify proof Metamath will report any proofs that are incorrect It is often useful to save the information that the Metamath program displays on the screen You can save everything that happens on the screen by opening a log file You may want to do this before you read in a database
196. ize e A f hypothesis tells Metamath the type or kind of its variable It is analogous to a variable declaration in a computer language that tells the compiler that a variable is an integer or a floating point number e The e hypothesis corresponds to what you would usually call a hy pothesis in ordinary mathematics Before an assertion a or p statement can be referenced in a proof all of its associated f and e hypotheses i e those e hypotheses that are active must be satisfied i e established by the proof The meaning of associated which we will call mandatory in Section 4 2 7 will become clear when we discuss scoping later 4 2 6 Assertions a and p Statements There are two types of assertions a statements axiomatic assertions and p statements provable assertions Their syntax is as follows 11A stronger version of a theorem like this would be the single formula P gt Q P implies Q from which the weaker version above follows by the rule of modus ponens in logic We are not discussing this stronger form here In the weaker form we are saying only that if we can prove P then we can prove Q In a logician s language if x is the only free variable in P and Q the stronger form is equivalent to Vz P gt Q for all z P implies Q whereas the weaker form is equivalent to VP VQ The stronger form implies the weaker but not vice versa To be precise the weaker form of the theorem is mo
197. ky but once we re there we have at our disposal a system that is less awkward to work with in formal proofs In technical terms that logicians understand we eliminate the cumbersome concepts of free variable bound variable and proper substitution as primitive notions These concepts are present in our system but are defined in terms of concepts expressed by the axioms and can be eliminated in principle In standard systems these concepts are really like additional implicit axioms that are somewhat complex and cannot be eliminated The traditional approach to logic wherein free variables and proper sub stitution is defined is also possible to do directly in the Metamath language However the notation tends to become awkward and there are disadvan tages for example extending the definition of a wff with a definition is awkward because the free variable and proper substitution concepts have to have their definitions also extended Our choice of axioms for set mm is to a certain extent a matter of style in an attempt to achieve overall simplicity but you should also be aware that the traditional approach is possible as well if you should choose it 45 Specifically it is a theorem of those systems of logic that assume non empty domains It is not a theorem of more general systems that include the empty domain in which nothing exists period Such systems are called free logics For a discussion of
198. l variables as well as local constants I have not decided upon the best way to do this or even if it should be done at all the main issue is whether the benefits offset greater confusion in learning the language In any case the rules for the present language were chosen to let existing databases be compatible with any future language extensions 16To make the loading faster the Metamath program does not automatically verify proofs when you read in a database unless you use the verify qualifier After a database has been read in you may use the verify proof command to verify proofs 4 3 THE ANATOMY OF A PROOF 115 mandatory and the others are optional You should review these concepts if necessary Each label in a proof must be either the label of a previous assertion a or p statement or the label of an active hypothesis e or f statement of the p statement containing the proof Hypothesis labels may reference both the mandatory and the optional hypotheses of the p statement The label sequence in a proof specifies a construction in reverse Pol ish notation RPN You may be familiar with RPN if you have used Hewlett Packard or similar hand held calculators In the calculator analogy a hypothesis label is like a number and an assertion label is like an operation On an RPN calculator an operation takes one or more previous numbers in an input sequence performs a calculation on them and replaces those numbers and itself with the
199. lation eliminating in definitions Edwards Robert E four color theorem effectively bound variable frame 108 effectively not free frames and scoping statements 114 element free logic empty domain free variable empty set 56 72 T75 106 170 free vs bound variable Frege Gottlob function in predicate calculus function value G del s incompleteness theorem Euclidean geometry kil 23 5 5 78 existential quantifier 3 65 69 gaps in proofs restricted Ghilbert existential uniqueness quantifier global statement 113 3 Goodstein R L expression 95 99 grave accent 119 120 in a formal system 163 Grothendieck Alexander extended frame group theory extended language Guillen Michael family Haken W Feferman Solomon Halmos Paul 21 Fermat s Last Theorem 13 Hamilton Alan G file inclusion Hardy G H file names Harrison John 20 Macintosh 41 123 Herrlich Horst Unix 41 i ee hierarchy 119 26 finitary proof 29 higher order logic INDEX Hilbert David 15 pes 29 30 Hindley J a Hofstadter Douglas a HOL 27 28 HTML generation Hume David 22 Huntington E V hypothesis hypothesis association hypothesis label iff image implication implicit axiom included file 93 individual metavariable individual variable inference inference rule infinite set infi
200. lem was later studied unsuccess fully by Tarski and his students and it remained an unsolved problem until a computer found the proof in 1996 For more information on the Robbins algebra problem see 67 1 3 THE USE OF COMPUTERS IN MATHEMATICS 27 How does Metamath relate to automated theorem provers A theorem prover is primary concerned with one theorem at a time perhaps tapping into a small database of known theorems whereas Metamath is more like a theorem archiving system storing both the theorem and its proof in a database for access and verification Metamath is one answer to what do you do with the output of a theorem prover and could be viewed as the next step in the process Automated theorem provers could be useful tools for helping develop its database although as of this writing there are no software translation tools that do this Note that very long automatically generated proofs can make your database fat and ugly and cause Metamath s proof verification to take a long time to run Unless you have a particularly good program that generates very concise proofs it might be best to consider the use of automatically generated proofs as a quick and dirty approach to be manually rewritten at some later date If you are interested in automatic theorem provers three well regarded programs are Isabell nor and orTER OTTER is also available on a disk included with the book Automated Reasoning Introduction and Appl
201. licit mathematical talent that frustrates many people from studying abstract mathematics The kind of mathematics that most people know limits them to the practical everyday usage of blindly manipulating numbers and formulas without any understanding of why those rules are correct nor any ability to go any further For example do you know why multiplying two negative numbers yields a positive number Starting with set theory you will also start off blindly manipulating symbols according to the rules we give you but with the advantage that these rules will allow you in principle to access all of mathematics not just a tiny part of it Of course concrete examples are often helpful in the learning process For example you can verify that that 2 3 3 2 by actually grouping objects and can easily see how it generalizes to y y x even though you might not be able to rigorously prove it Similarly in set theory it can be helpful to understand how the axioms of set theory apply to and are correct for small finite collections of objects You should be aware that in The statement x is a natural number is formally expressed as x w where stylized epsilon means is in or is an element of and w omega means the set of natural numbers When xz w is completely expanded in terms of the primitive symbols of set theory the result isa Vz 7AVw ze wrr1we2rz gt zer S Vz 4 Vw wezowesz oVw
202. literature 34A Treatise of Human Nature as quoted in 14 p 267 35 Tq p 269 1 2 COMPUTERS AND MATHEMATICIANS 23 e There have been thousands of purported proofs of Fermat s Last The orem no integer solutions exist to z y z for n gt 2 by amateurs cranks and well regarded mathematicians 56 p 5 Fer mat wrote a note in his copy of Bachet s Diophantus that he found a truly marvelous proof of this theorem but this margin is too nar row to contain it 29 p 507 A recent much publicized proof by Yoichi Miyaoka was shown to be incorrect Science News April 9 1988 p 230 The theorem was finally proved by Andrew Wiles Sci ence News July 3 1993 p 5 but it initially had some gaps and took over a year after its announcement to be checked thoroughly by ex perts On Oct 25 1994 Wiles announced that the last gap found in his proof had been filled in e In 1882 M Pasch discovered that an axiom was omitted from Euclid s formulation of geometry without it the proofs of important theorems of Euclid are not valid Pasch s axiom states that a line that intersects one side of a triangle must also intersect another side provided that it does not touch any of the triangle s vertices The omission of Pasch s axiom went unnoticed for 2000 years 13 p 160 in spite of one presumes the thousands of students instructors and mathematicians who studied Euclid e The first publish
203. ll ordinal numbers An ordinal number is a set that satisfies the ordinal predicate Definition 7 11 of Takeuti and Zaring p 38 df on a On 2 Ord z df on a On x Ord x Define the limit ordinal predicate which is true for a non empty ordinal that is not a successor i e that is the union of itself Compare Bell and Machover p 471 and Exercise 1 p 42 of Takeuti and Zaring df lim a Lim AG Ord AAA A AA UA df lim a Lim A lt gt Ord A A A U A Define the successor of a class Definition 7 22 of Takeuti and Zaring p 41 Our definition is a generalization to classes although it is meaningless when classes are proper df suc a F suc A AU A df suc a suc A C Au A Define the class of natural numbers Compare Bell and Machover p 471 df om a Fw 2 Ord rAVy Lim y gt rey df om a om x Ord x A y Lim y gt xe y Define the cross product of two classes Definition 9 11 of Quine p 64 df xp a Ax B a2 y t EAAyEB df xp a AX B lt x y gt xe A ye B Define the domain of a class Definition 6 5 1 of Takeuti and Zaring p 24 df dm a domA ax dy xz y EA df dm a dom A x E y lt x y gt e A S 3 4 A HIERARCHY OF DEFINITIONS 75 Define the range of a class Definition 6 5 2 of Takeuti and Zaring p 24 df rn a F ranA y da az y EA df rn a ran A 4 y E x lt x y gt e A S
204. ll take a look at a very simple example from formal number theory This example is taken from Mendelson 36 p 123 We will look at a small subset of this theory namely that part needed for the first number theory theorem proved in 36 First we will look at a standard formal proof for the example we have picked then we will look at the Metamath version If you have never been exposed to formal proofs the notation may seem to be such overkill to ex press such simple notions that you may wonder if you are missing something You aren t The concepts involved are in fact very simple and a detailed breakdown in this fashion is necessary to express the proof in a way that can be verified mechanically And as you will see Metamath breaks the proof down into even finer pieces so that the mechanical verification process can be about as simple as possible Before we can introduce the axioms of the theory we must define the syntax rules for forming legal expressions combinations of symbols with which those axioms can be used The number 0 is a term and if t and r are terms so is t r Here t and r are metavariables ranging over terms they themselves do not appear as symbols in an actual term Some examples of actual terms are 0 0 and 0 0 0 Note that our theory describes only the number zero and sums of zeroes Of course not much can be done with such a trivial theory but remember that we have picked a very small subset
205. lowed by zero or more active math symbols followed by A p statement consists of a label followed by p followed by an active constant followed by zero or more active math symbols followed by followed by a sequence of labels followed by An assertion is a a or p statement A f e or d statement is active from the place it occurs until the end of the block it occurs in A a or p statement is active from the place it occurs through the end of the database There may not be two active f statements containing the same vari able Each variable in a e a or p statement must exist in an active f statement Each label token must be unique No label token may match any math symbol token The set of mandatory variables associated with an assertion is the set of zero or more variables in the assertion and in any active e statements The possibly empty set of mandatory hypotheses is the set of all active f statements containing mandatory variables together with all active e statements The set of mandatory d statements associated with an assertion are those active d statements whose variables are both among the assertion s mandatory variables 4The rules for redeclaration may become more general in the future see footnote on is requirement can greatly simplify the unification algorithm substitution calcu lation required by proof verification 6 Added June 24 2006 This restriction did not exist
206. lton Logic for Mathematicians Cambridge University Press Cambridge revised edition 1988 QA9 H298 John Robert Harrison Metatheory and reflection in theorem proving A survey and critique Technical Report CRC 053 SRI Cambridge Millers Yard Cambridge UK 1995 Available on the Web as http w0w cl cam ac uk users jrh papers reflect html John Robert Harrison Theorem proving with the real numbers Tech nical Report 408 University of Cambridge Computer Laboratory New Museums Site Pembroke Street Cambridge CB2 3QG UK 1996 Au thor s PhD thesis available on the Web at http www cl cam ac uk users jrh papers thesis html BIBLIOGRAPHY 183 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Horst Herrlich and George E Strecker Category Theory An Introduc tion Allyn and Bacon Inc Boston 1973 QA169 H567 J Roger Hindley and David Meredith Principal type schemes and condensed detachment The Journal of Symbolic Logic 55 90 105 1990 QA J87 Douglas R Hofstadter Godel Escher Bach Basic Books Inc New York 1979 QA9 H63 1980 D Kalish and R Montague On Tarski s formalization of predicate logic with identity Archiv f r Mathematische Logik und Grundlagenfor schung 7 81 101 1965 QA A673 J A Kalman Condensed detachment as a rule of inference Studia Logica 42 4 443 451 1983 B18 P6 S933 Morris Kline Mathematical Thou
207. m You may find it instructive to compare them Appendix D The MIU System The following is a listing of the file miu mm It is self explanatory The MIU system A simple formal system Note This formal system is unusual in that it allows empty wffs To work with a proof you must type SET EMPTY_SUBSTITUTION ON before using the PROVE command By default this is OFF in order to reduce the number of ambiguous unification possibilities that have to be selected during the construction of a proof Hofstadter s MIU system is a simple example of a formal system that illustrates some concepts of Metamath See Douglas R Hofstadter _Goedel Escher Bach An Eternal Golden Braid_ Vintage Books New York 1979 pp 33ff for a description of the MIU system The system has 3 constant symbols M I and U The sole axiom of the system is MI There are 4 rules Rule I If you possess a string whose last letter is I you can add on a U at the end Rule II Suppose you have Mx Then you may add Mxx to your collection Rule III If III occurs in one of the strings in your collection you may make a new string with U in place of III Rule IV If UU occurs inside one of your strings you can drop it Unfortunately Rules III and IV do not have unique results 177 178 APPENDIX D THE MIU SYSTEM strings could have more than one occurrence of III or UU This requires that we introduce the concept of an MIU well formed for
208. mber no_distinct and improve all depth number no_distinct 146 CHAPTER 5 THE METAMATH PROGRAM These commands available in the Proof Assistant only try to find proofs automatically for unknown steps whose symbol sequences are completely known They are primarily useful for filling in proofs of f hypotheses The search will be restricted to statements having no e hypotheses Note If memory is limited improve all on a large proof may over flow memory If you use set unification_timeout 1 before improve all there will usually be sufficient improvement to easily recover and completely improve the proof later on a larger computer Warning Once memory has overflowed there is no recovery If in doubt save the intermediate proof save new_proof then write source before improve all If last is specified instead of step number the last step that is shown by show new_proof unknown will be used If first is specified instead of step number the first step that is shown by show new_proof unknown will be used If step is zero or negative the stepth from last unknown step as shown by show new_proof unknown will be used improve 1 will use the penul timate unknown step improve 2 label the antepenultimate and improve 0 is the same as improve last Optional command qualifier depth number This qualifier will cause the search to include state ments with e hypotheses but no new variables in the e hypotheses pro vided
209. mp a t t MM PA gt assign 8 al To undo the assignment DELETE STEP 11 and INITIALIZE UNIFY if needed T min t 0 t MM PA gt assign 7 a2 To undo the assignment DELETE STEP 8 and INITIALIZE UNIFY if needed MM PA gt show new_proof lemmon 4 a2 a l t 0 t 2 4 YOUR FIRST PROOF 8 12 13 8 12 mp 14 4 13 mp a2 al a a a a l l I l t Pan ct o Oo wv t 0 t t t 5l gt t 0 t gt Er a a A gt t t Now all temporary variables and unknown steps have been eliminated from the essential part of the proof When this is achieved the Proof Assistant can usually figure out the rest of the proof automatically Note that the improve command can occasionally be useful for filling in essential steps as well but it only tries to make use of statements that introduce no new variables in their hypotheses which is not the case for mp Also it will not try to improve steps containing temporary variables Let s look at the complete proof then run the improve command then look at it again MM PA gt show new_proof lemmon all wff t 0 t 1 OANDOABWHD erer Ne oO 13 14 a2 5 3 7 a2 9 10 11 al 5 6 152 8 4 12 mp 13 mp a a a a a MM PA gt improve all rrrrrrrr pe st proof proof proof proof proof proof proof proof proof eps 1 of of of o
210. mputer Language The study of mathematics is apt to commence in disappoint ment We are told that by its aid the stars are weighted and the billions of molecules in a drop of water are counted Yet like the ghost of Hamlet s father this great science eludes the efforts of our mental weapons to grasp it ALFRED NORTH WHITEHEAD 1 1 1 Is Mathematics User Friendly Suppose you have no formal training in abstract mathematics But popular books you ve read offer tempting glimpses of this world filled with profound ideas that have stirred the human spirit You are not satisfied with the informal watered down descriptions you ve read but feel it is important to grasp the underlying mathematics itself to understand its true meaning It s not practical to go back to school to learn it though you don t want to dedicate years of your life to it There are many important things in life and you have to set priorities for what s important to you What would happen if you tried to pursue it on your own in your spare time After all you were able to learn a computer programming language such as Pascal on your own without too much difficulty even though you had no formal training in computers You don t claim to be an expert in software design but you can write a passable program when necessary to suit your needs Even more important you know that you can look at anyone else s Pascal program no matter how complex and with
211. ms for essentially all of mathematics Wonder at them and stare at them in awe Put a copy in your wallet and you will carry in your pocket the encoding for all theorems ever proved and that ever will be proved from the most mundane to the most profound 3 4 A Hierarchy of Definitions The axioms in the previous section in principle embody everything that can be done within standard mathematics However it is impractical to accom plish very much by using them directly for even simple concepts from a human perspective can involve extremely long incomprehensible formulas Mathematics is made practical by introducing definitions Definitions usu ally introduce new symbols or at least new relationships among existing symbols to abbreviate more complex formulas An important requirement for a definition is that there exist a straightforward algorithmic method for eliminating the abbreviation by expanding it into the more primitive symbol string that it represents Some important definitions included in the file set mm are listed in this section for reference and also to give you a feel for why something like w the set of natural numbers 0 1 2 becomes very complicated when completely expanded into primitive symbols What is the motivation for definitions aside from allowing complicated expressions to be expressed more simply In the case of w one goal is to provide a basis for the theory of natural numbers Before set theory was i
212. mula or wff which allows us to construct unique symbol sequences to which Rules III and IV can be applied First we declare the constant symbols of the language Note that we need two symbols to distinguish the assertion that a sequence is a wff from the assertion that it is a theorem we have arbitrarily chosen wff and c MIU wff Declare constants Next we declare some variables v x y Throughout our theory we shall assume that these variables represent wffs wx f wff x wy f wff y Define MIU wffs We allow the empty sequence to be a wff The empty sequence is a wff we a wff M after any wff is a wff wM a wff x M I after any wff is a wff wl a wff x I U after any wff is a wff wU a wif x U Assert the axiom ax a MI Assert the rules Ia e x I Given any theorem ending with I it remains a theorem if U is added after it We distinguish the label I_ from the math symbol I to conform to the 24 Jun 2006 Metamath spec I a x IU 179 IIa e Mx Given any theorem starting with M it remains a theorem if the part after the M is added again after it II a Mxx IIIa e x IIL y Given any theorem with III in the middle it remains a theorem if the III is replaced with U III a l xUy IVa e x UUy
213. n as the real axioms and Metamath does not distinguish between the two The distinction is made by you i e by the way in which you interpret the notation you have chosen to express these two kinds of axioms The Metamath language breaks down mathematical proofs into tiny pieces much more so than in ordinary formal proofs If a single step in volves the substitution of a complex term for one of its variables Metamath must see this single step broken down into many small steps This fine grained breakdown is what gives Metamath generality and flexibility as it lets it not be limited to any particular mathematical notation Metamath s proof notation is not in itself intended to be read by hu mans but rather is in a compact format intended for a machine The Meta math program will convert this notation to a form you can understand using the show proof command You can tell the program what level of detail of the proof you want to look at You may want to look at just the logical inference steps that correspond to ordinary formal proof steps or you may want to see the fine grained steps that prove that an expression is a term Here without further ado is our example converted to the Metamath language Declare the constant symbols we will use c 0 gt term wff Declare the metavariables we will use vtrsPQ Specify properties of the metavariables tt f term t 5Some authors make this im
214. n opportunity to write source to permanently save the changes In Proof Assistant mode the exit command will return to the MM gt prompt If there were changes to the proof you will be given an oppor tunity to save new_proof The quit command is a synonym for exit Optional qualifier force Do not prompt if changes were not saved This qualifier is useful in submit command files Section 5 2 4 to ensure predictable behavior 5 2 2 open log Command Syntax open log file name This command will open a log file that will store everything you see on the screen It is useful to help recovery from a mistake in a long Proof Assistant session or to document bugs The log file can be closed with close log It will automatically be closed upon exiting Metamath 132 CHAPTER 5 THE METAMATH PROGRAM 5 2 3 close log Command Syntax close log The close log command closes a log file if one is open See also open log 5 2 4 submit Command Syntax submit filename This command causes further command lines to be taken from the spec ified file Note that any line beginning with an exclamation point is treated as a comment i e ignored Also note that the scrolling of the screen output is continuous so you may want to open a log file see open log to record the results that fly by on the screen After the lines in the file are exhausted Metamath returns to its normal user interface mode Currently the submit command is not recursi
215. n this case the wrapping of the information Messages may sometimes seem somewhat unnatural however In TpRxX there is normally a maximum of 61 characters per line with typewriter font The examples in this book were produced with 61 characters per line 5 2 9 set height Command Syntax set height number Metamath assumes your screen height is 24 lines of characters If your screen is taller or shorter this command lets you to change the number of lines at which the display pauses and prompts you to continue 5 2 10 beep Command Syntax beep This command will produce a beep By typing it ahead after a long running command has started it will alert you that the command is finished For convenience b is an abbreviation for beep Note If b is typed at the MM gt prompt immediately after the end of a multiple page display paged with Press lt return gt for more prompts then the b will back up to the previous page rather than perform the beep command In that case you must type the unabbreviated beep form of the command 5 2 11 more Command Syntax more filename This command will display the contents of an ASCII file on your screen This command is provided for convenience but is not very powerful See Section 5 2 12 to invoke your operating system s command to do this such as the more command in Unix 5 2 12 Operating System Commands A line enclosed in single or double quotes will be executed by your compute
216. nal command qualifier no_header This qualifier prevents a standard TX header and trailer from being included with the output T px code 5 7 2 close tex Command Syntax close tex 148 CHAPTER 5 THE METAMATH PROGRAM This command writes a trailer to any BT X file that was opened with open tex unless no_header was used with open tex and closes the MTX file 5 8 Creating HTML Output The ability to produce HTML web pages is new in Metamath version 0 07 30 To create an HTML output file for a a or p statement use show statement label html The output file will be named label htm1 When label has wildcard characters all statements with matching labels will have HTML files produced for them Also when label has a wildcard character two additional files mmdefinitions html and mmascii html will be produced To produce only these two additional files you can use which will not match any statement label in place of label There are three other qualifiers for show statement that also generate HTML code These are alt_html brief_html and brief_alt_html and are described in the next section A statement s comment can include a special notation that provides a certain amount of control the HTML version of the comment See Sec tion 4 4 1 p for the comment markup features The write theorem_list and write bibliography commands which are described below provide as a side effect complete error checking for
217. nd in with the mathematics being described not distract from it like punctuation marks 4 2 1 User Defined Tokens As you may have noticed all keywords begin with the character This mundane monetary symbol is not ordinarily used in higher mathematics outside of grant proposals so we have appropriated it to distinguish the Metamath keywords from ordinary mathematical symbols The character is thus considered special and may not be used as a character in a user defined token All tokens and keywords are case sensitive for example n is considered to be a different character from N Case sensitivity makes the available ASCII character set as rich as possible Math Symbol Tokens Math symbols are tokens used to represent the symbols that appear in or dinary mathematical formulas They may consist of any combination of the 93 printable ASCII characters other than Some examples are x 1 07 amp and bounded For readability it is best to try to make these look as similar to actual mathematical symbols as possible within the constraints of the ASCII character set in order to make the resulting mathematical ex pressions more readable In the Metamath language you express ordinary mathematical formu las and statements as sequences of math symbols such as 2 2 4 five 98 CHAPTER 4 THE METAMATH LANGUAGE symbols all constants They may even be English sentences as in E is closed and bounded five symbols here
218. ngerous definition df cleq 1 e F Va r eEyouez gt y z df cleq a A BHVa rEeAoxreB df cleq 1 e A x xe y lt gt xe z gt y z df cleq a l A B lt gt A x xe A lt gt xe B Define the membership connective between classes Theorem 6 3 of Quine p 41 which we adopt as a definition Note that it extends the use of the existing membership symbol but unlike df cleq it does not extend the set of valid wffs of logic when the class metavariables are replaced with set variables df clel a AC BH dr t AArEB df clel a Ae B lt gt E x x A xe B Define inequality df ne a k AABUH AA B df ne a A B lt gt A B Define restricted universal quantification Enderton p 22 df ral a F VreAypoVa reEeAy df ral a A xe A ph lt gt A x xe A gt ph Define restricted existential quantification Enderton p 22 df rex a dre Apo dr reArNy df rex a E xe A ph lt gt E x xe A ph 72 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED Define the universal class Definition 5 20 p 21 of Takeuti and Zaring df v a V a x 2 df v a V x x x Define the subclass relationship between two classes called the subset rela tion if the classes are sets i e are not proper Definition 5 9 of Takeuti and Zaring p 17 df ss a k ACBoVa rxeE Ave B df ss a A _B lt gt A x xe A gt xe B
219. ngly enough be derived from a relatively small set of axioms or first principles The development of these axioms is among the most impressive and important accomplishments of mathematics in the 20th century Ultimately these axioms can be broken down into a set of rules for manipulating symbols that any technically oriented person can follow We will not spend much time trying to convey a deep higher level under standing of the meaning of the axioms This kind of understanding requires some mathematical sophistication as well as an understanding of the phi losophy underlying the foundations of mathematics and typically develops over time as you work with mathematics Our goal instead is to give you the immediate ability to follow how theorems are derived from the axioms and from other theorems This will be similar to learning the syntax of a computer language which lets you follow the details in a program but does not necessarily give you the ability to write non trivial programs on your own an ability that comes with practice For now don t be alarmed by abstract sounding names of the axioms just focus on the rules for manip ulating the symbols which follow the simple conventions of the Metamath language 18 p 31 55 56 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED The axioms that underlie all of standard mathematics consist of axioms of logic and axioms of set theory The axioms of logic are divided into two subcategories pro
220. nity infix connective informal proof integer intersection intuitionism Isabelle Kalman J A Bl Kempe A B 23 keyword Kline Morris Koch K Kronecker Leopold Kuratowski Kazimierz label 39 42 45 93 96 98 103 105 07 108 115 label declaration 98 label mode 120 121 label reference 98 a m label sequence 98 labels in set mm Landau Edmund 191 YTX E B3 E3 S I 3 LAT characters per line 133 BTEX definitions 155 LCF 28 Le niewski S Leblanc Hugues Lejewski Czeslaw Lemmon style proof length of a sequence 162 Levien Raph limit ordinal local label 90 local variable logic 2 logical equivalence SAAE logical hypothesis in a formal system 164 logical AND A 65 logical OR V Lounesto Pertti Mac Lane Saunders Macintosh file names 41 MACSYMA 3 mandatory d statement mandatory disjoint variable restriction in a formal system mandatory hypothesis 39 42 oa LY LT LT TT TL in a formal system 164 mandatory variable mandatory variable type hypothesis in a formal system 164 Maple mapping markup notation 119 E mi math mode 119 math eae Be w 60 p3 96 af rey Mathematica 3 24 Mathematica and proofs mathematical induction 192 Mathias Adrian R D Megill Norman 17 AAE member Mendelson Ellio
221. nly through much practice and hard work Only the few who complete the painful learning experience can join the small in group of pure mathematicians The informal language 1 E3 p 40 2 CHAPTER 1 INTRODUCTION effectively cuts off the true nature of their knowledge from most everyone else Metamath makes abstract mathematics more concrete It allows a com puter to keep track of the complexity associated with each word or symbol with absolute rigor You can explore this complexity at your leisure to whatever degree you desire Whether or not you believe that concepts such as infinity actually exist outside of the mind Metamath lets you get to the foundation for what s really being said Its language is simple enough so that you don t have to rely on the authority of experts but can verify the results yourself step by step If you want to attempt to derive your own results Metamath will not let you make a mistake in reasoning Metamath is the name of a mathematical computer language that de scribes formal mathematical systems and expresses proofs of theorems in those systems Such a language is called a metalanguage by mathemati cians Metamath is also the name of a computer program that verifies proofs expressed in the language The Metamath program does not have the built in ability to make logical inferences it just makes a series of sym bol substitutions according to instructions given to it in a proof and v
222. not include line breaks A token string or latex string may in clude a quotation mark that matches the enclosing quotes by repeating the quotation mark twice for example the token strings wariu Ic d e fe g h specify the tokens a b c d e f and g h respectively Finally a long latez string may be broken up into multiple quote enclosed strings joined by in order to fit them on several lines thus ap cq ref is the same as abcdef The t comment may also contain htmldef statements althtmldef statements and some other types of statements related to the generation of HTML pages for the Metamath web site The syntax for these is not yet final and for current information you should consult help html in the Metamath program Importantly whenever you add a latexdef statement you should also add a new htmldef statement and a new althtmldef statement to keep all symbol definitions up to date 157 Symbol dom ran Rel Fun Fn onto II onto rec to Er Token dom ran Rel Fun Fn 1 1 gt onto gt 1 1 onto gt rec Er Symbol UI Tr Po Or Fr We Ord On Lim suc Token e P Tr Po Or Fr We Ord On Lim suc om X Symbol wif set class Token wif ph ps ch th ta et ze set
223. nother exit will then get you out of Metamath 5 6 1 prove Command Syntax prove label This command will enter the Proof Assistant which will allow you to cre ate or edit the proof of the specified statement The command line prompt will change from MM gt to MM PA gt Note In the present version of Metamath 0 07 30 the Proof Assistant does not verify that d restrictions are met as a proof is being built After you have completed a proof you should type save new_proof followed by verify proof label where label is the statement you are proving with the prove command to verify the d restrictions See also exit 5 6 2 set unification timeout Command Syntax set unification_timeout number This command is available outside the Proof Assistant but affects the Proof Assistant only Sometimes the Proof Assistant will inform you that a unification time out occurred This may happen when you try to unify formulas with many temporary variables 1 2 etc since the time to compute all possible unifications may grow exponentially with the number of variables If you want Metamath to try harder and you re willing to wait longer you may increase this parameter show settings will show you the current value 142 CHAPTER 5 THE METAMATH PROGRAM 5 6 3 set empty_substitution Command Syntax set empty_substitution on or set empty_substitution off This command is available outside the Proof Assistant but affects the Proof Assi
224. nother set 24 CHAPTER 1 INTRODUCTION Mathematician H S M Coxeter has doubts 13 p 58 I have a feeling that is an untidy kind of use of the computers and the more you correspond with Haken and Appel the more shaky you seem to be e Many false proofs of the Poincar conjecture have been proposed over the years This conjecture states that any object that mathemat ically behaves like a three dimensional sphere is a three dimensional sphere topologically regardless of how it is distorted In March 1986 mathematicians Colin Rourke and Eduardo R go caused a stir in the mathematical community by announcing that they had found a proof in November of that year the proof was found to be false 46 p 218 It was finally proved in 2003 by Grigory Perelman 58 Many counterexamples to theorems in recent mathematical literature related to Clifford algebras have been found by Pertti Lounesto who passed away in 2002 See the web page http mathforum org library view One of the purposes of Metamath is to allow proofs to be expressed with absolute precision Developing a proof in the Metamath language can be challenging because Metamath will not permit even the tiniest mistake But once the proof is created its correctness can be trusted immediately without having to depend on the process of peer review for confirmation 1 3 The Use of Computers in Mathematics 1 3 1 Computer Algebra Systems For the most part
225. nput from one or more source files which contain characters expressed in the standard ASCII American Standard for Coded Information Interchange code for comput ers A source file consists of a series of tokens which are strings of printable characters from the set of 94 shown on p separated by white space spaces tabs carriage returns line feeds and form feeds Any string con sisting only of these characters is treated the same as a single space The printable characters that Metamath recognizes are the 94 characters on stan dard Ascit keyboards Metamath has the ability to join several files together to form its input Section 4 4 3 We call the aggregate contents of all the files after they have been joined together a database to distinguish it from an individual source file The tokens in a database consist of keywords which are built into the language together with two kinds of user defined tokens called labels and math symbols Often we will simply say symbol instead of math symbol for brevity The set of basic keywords is c v e f d a p and This is the complete set of syntactical elements of what we call the basic language of Metamath and with them you can express all of the mathematics that were intended by the design of Metamath You should make it a point to become very familiar with them Table 4 1 lists the basic keywords along with a brief description of their functions For now this descript
226. ns itself with statements that can be inter preted as either true or false Some examples of statements outside of mathematics that are either true or false are It is raining today and The United States has a female president In mathematics as we mentioned statements are really formulas In propositional calculus we don t care what the statements are We also treat a logical combination of statements such as It is raining today and the United States has a female president no differently from a single statement Statements and their combinations are called well formed formulas wffs We define wffs only in terms of other wffs and don t define what a starting wff is As is common practice in the literature we use Greek letters to represent wfts Specifically suppose y and w are wffs Then the combinations y gt w p implies y also read if y then Y and n not p are also wffs The three axioms of propositional calculus are all wffs of the following form g gt Y gt p Y Y gt x gt y gt y gt gt x e gt 4 gt Y gt p There are an infinite number of axioms one for each possible wff of the above form For this reason axioms such as the above are often called 3A remarkable result of C A Meredith squeezes these three axioms into the single axiom p Y gt 4x gt 78 x gt 7 gt r gt p gt 0 gt p B7 which is believed to be the
227. nstants respectively The same math symbol may not occur twice in a given v or c statement A math symbol becomes active when declared and stays active until the end of the block in which it is declared A variable may not be declared a second time while it is active but it may be declared again as a variable but 94 CHAPTER 4 THE METAMATH LANGUAGE not as a constant after it becomes inactive A constant must be declared in the outermost block and may not be declared a second timeff A f statement consists of a label followed by f followed by an active constant followed by an active variable followed by A e statement consists of a label followed by e followed by an active constant followed zero or more active math symbols followed by A hypothesis is a f or e statement A simple d statement consists of d followed by two different active variables followed by A compound d statement consists of d followed by three or more variables all different followed by The order of the variables in a d statement is unimportant A compound d statement is equivalent to a set of simple d statements one for each possible pair of variables occurring in the compound d statement Henceforth in this specification we shall assume all d statements are simple A d statement is also called a disjoint or distinct variable restriction A a statement consists of a label followed by a followed by an active constant fol
228. nt to clean up delete these n files 5 3 1 read Command Syntax read file name verify This command will read in a Metamath language source file and any included files Normally it will be the first thing you do when entering Metamath Statement syntax is checked but proof syntax is not checked Note that the file name may be enclosed in single or double quotes this is useful if the file name contains slashes as might be the case under Unix If you are getting an Expected VERIFY error when trying to read a Unix file name with slashes you probably haven t quoted it If you are prompted for the file name by pressing return after read you should not put quotes around it even if it is a Unix file name with slashes Optional command qualifier verify Verify all proofs as the database is read in This qualifier will slow down reading in the file See verify proof for more information on file error checking See also erase 5 3 2 write source Command Syntax write source filename clean This command will write the contents of a Metamath database into a file Note The present version of Metamath 0 07 30 will not split the 5 4 SHOWING STATUS AND STATEMENTS 135 database into its constituent source files included with and keywords A future version is planned to properly separate all constituent files Optional command qualifier primarily intended to assist web site up dates clean Suppresses delet
229. nvented a set of axioms for arithmetic called Peano s postulates was devised and shown to have the properties one expects for natural numbers Now anyone can postulate a set of axioms but if the axioms are inconsistent contradictions can be derived from them Once a contradiction is derived anything can be trivially proved including all the facts of arithmetic and their negations To ensure that an axiom system is at least as reliable as the axioms for set theory we can define sets and operations on those sets that satisfy the new axioms In the set mm Metamath database we prove that the elements of w satisfy Peano s postulates and it s a long and hard journey to get there directly from the axioms of set theory But the result is 3 4 A HIERARCHY OF DEFINITIONS 67 confidence in the foundations of arithmetic And there is another advantage we now have all the tools of set theory at our disposal for manipulating objects that obey the axioms for arithmetic What are the criteria we use for definitions First and of utmost impor tance the definition should not be creative that is it should not allow an expression that previously qualified as a wff but was not provable to become provable Second the definition should be eliminable that is there should exist an algorithmic method for proving any expression using the definition into a logically equivalent expression that previously qualified as a wff In almost all cases below defini
230. o check whether a formula is a tautology is called the truth table method and it works like this A wf p gt y is false whenever is true and w is false Otherwise it is true A wff 7y is false whenever vy is true and false otherwise To verify a tautology such as yp gt y gt vy you break it down into sub wffs and construct a truth table that accounts for all possible combinations of true and false assigned to the wff metavariables lell y gt e gt t gt o TIT T T TIF T T F T F T FIF T T If all entries in the last column are true the formula is a tautology Now the truth table method doesn t tell you how to prove the tautology from the axioms but only that a proof exists Finding an actual proof especially one that is short and elegant can be challenging Methods do exist for automatically generating proofs in propositional calculus but the proofs that result can sometimes be very long In the Metamath set mm database most or all proofs were created manually 3 2 2 Predicate Calculus Predicate calculus introduces the concept of individual variables which we will usually just call variables These will always represent sets when 60 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED we get to set theory There are also three new symbols V and read for all equals and is an element of respectively We will represent variables with the let
231. o valued propositional calculus The Journal of Computing Systems 3 155 164 1953 David Meredith In memoriam Carew Arthur Meredith 1904 1976 Notre Dame Journal of Formal Logic 18 513 516 1977 QA N914 J Donald Monk Substitutionless predicate logic with identity Archiv fiir Mathematische Logik und Grundlagenforschung 7 103 121 1965 A W Moore The Infinite Routledge New York 1989 BD411 M59 James R Munkres Topology A First Course Prentice Hall Inc Englewood Cliffs New Jersey 1975 QA611 M82 E Z Nemesszeghy and E A Nemesszeghy On strongly creative defini tions A reply to V F Rickey Logique et Analyse N S 20 111 115 1977 BC L832 I N meti Algebraizations of quantifier logics an overview Ver sion 11 4 preprint Mathematical Institute Budapest 1994 A short ened version without proofs appeared in Algebraizations of quanti fier logics an introductory overview Studia Logica 50 485 569 1991 B18 P6 S933 M Pavi i A new axiomatization of unified quantum logic Interna tional Journal of Theoretical Physics 31 1753 1766 1992 QC 1626 Roger Penrose The Emperor s New Mind Oxford University Press New York 1989 Q335 P415 Ivars Peterson The Mathematical Tourist W H Freeman and Com pany New York 1988 QA93 P475 Jeremy George Peterson An automatic theorem prover for substitution and detachment systems Notre Dame Journal of Formal Logic 19
232. of complete number theory for our example The important thing for you to focus on is our definitions that describe how symbols are combined to form valid expressions and not on the content or meaning of those expressions If and r are terms an expression of the form t r is a wff well formed formula and if P and Q are wits so is P Q which means P implies Q or if P then Q Here P then Q are metavariables ranging over wffs Examples of actual wffs are 0 0 3HyperText Markup Language 4To keep the example simple we have changed the formalism slightly and what we call axioms are strictly speaking theorems in 36 2 2 YOUR FIRST FORMAL SYSTEM 35 0 0 0 0 0 gt 0 0 0 and 0 0 gt 0 0 gt 0 0 0 Our notation makes use of more parentheses than are customary but the elimination of ambiguity this way simplifies our example by avoiding the need to define operator precedence The axioms of our theory are all wffs of the following form where t r and s are any terms Al t r gt t s gt r s A2 t 0 t Note that there are an infinite number of axioms since there are an infinite number of possible terms Al and A2 are properly called axiom schemes but we will refer to them as axioms for brevity An axiom is a theorem and if P and P gt Q are theorems where P and Q are wffs then Q is also a theorem The second part of this definition is called the modus ponens M
233. of id1 essential lemmon renumber 1 ax 2 a I ph gt ph gt ph gt ph gt ph gt ph gt ph gt ph gt ph 2 ax 1 a l ph gt ph gt ph gt ph 3 1 2 ax mp a ph gt ph gt ph gt ph gt ph 4 ax 1 a ph gt ph gt ph 5 3 4 ax mp a ph gt ph If you have read Section you ll know how to interpret this proof Step 2 for example is an application of axiom ax 1 This proof is identical to the one in Hamilton s Logic for Mathematicians I9 p 32 You may want to look at what substitutions are made into ax 1 to arrive at step 2 The command to do this needs to know the real step number so we ll display the proof again without the renumber qualifier MM gt show proof id1 lemmon essential 18 ax 2 a l ph gt ph gt ph gt ph gt ph gt ph gt ph gt ph gt ph 21 ax 1 a l ph gt ph gt ph gt ph 22 18 21 ax mp a ph gt ph gt ph gt ph gt ph 25 ax 1 a ph gt ph gt ph 26 22 25 ax mp a ph gt ph The real step number is 21 Let s look at its details MM gt show proof idi detailed_step 21 Proof step 21 min ax 1 a ph gt ph gt ph gt ph This step assigns source ax 1 a to target min e The source assertion requires the hypotheses wph f step 19 and wp
234. of is saved in the compressed format which re duces storage requirements for a database See Appendix B 5 7 Creating BTRX Output The show statement and show proof commands each have a special tex command qualifier that produces ATEX output The show statement com mand also has the simple_tex qualifier for output that is easier to edit by hand Before you can use them you must open a ATEX file to which to send their output A typical complete session will use this sequence of Metamath commands read set mm open tex example tex show statement ali tex show proof ali lemmon renumber tex show statement uneq2 tex show proof uneq2 lemmon renumber tex close tex See Section 4 4 1 for information on comment markup and Appendix A for information on how math symbol translation is specified To format and print the TRX source you will need the TeX program which is standard on most Linux installations and available for Windows On Linux in order to create a PDF file you will typically type at the shell prompt pdflatex example tex 5 7 1 open tex Command Syntax open tex file name no_header This command opens a file for writing BT X source and writes a PT EX header to the file BTEX source can be written with the show proof show new_proof and show statement commands using the tex qualifier The mapping to ATEX symbols is defined in a special comment contain ing a t token described in Appendix A Optio
235. oms with OTTER they must be restated in a way that eliminates the need for dummy variables See the Comment on p 105 AC 4 28 CHAPTER 1 INTRODUCTION pure mathematics is primarily focused on automated theorem provers not verifiers again with the ultimate goal of aiding the creation of new math ematics Automated theorem provers are usually concerned with attacking one theorem at time rather than making a large organized database easily available to the user Metamath is one way to help close this gap Besides Metamath there are several other on going projects with the goal of formalizing mathematics into computer verifiable databases One such project is QED and several mathematicians are currently working to agree on the requirements for a universal language Information on this project is available at One specific proof verification language is Mizar which can display its proofs in the informal language that mathematicians are accustomed to Information on the Mizar language is available http mizar org Other higher level proof verification languages are LCF and HOL a good overview of these and others is given in 20 All of these languages are fundamentally different from Metamath in that much of the mathemati cal foundational knowledge is embedded in the underlying proof verification program rather than placed directly in the database that is being verified For the working mathematician these languages
236. on Metamath is what might be thought of as a compile only language rather than an algorithmic or Turing machine language Pascal C Prolog Mathematica etc In other words a program database written in the Metamath language doesn t do anything it merely exhibits mathematical knowledge and permits this knowledge to be verified as being correct A program in an algorithmic language can potentially have hidden bugs as well as pos sibly being hard to understand But each token in a Metamath database PREFACE xi must be consistent with the database s earlier contents according to simple fixed rules and if a database is syntactically correct P then the mathemat ical content is correct with absolute certainty or at least to the certainty of the verification program which is relatively simple The only bugs that can exist are in the statement of the axioms for example if the ax ioms are inconsistent a famous problem shown to be unsolvable by G del s incompleteness theorem Metamath doesn t prove theorems automatically but is designed to ver ify proofs that you supply to it Metamath is completely general and has no built in preconceived notions about your formal system its logic or its syntax but the price for its generality is that it does not lend itself well to automated proofs in its most general form In principle it could accept translated proofs from other more specific theorem proving programs
237. on of Enderton p 19 df tp a F A B C A B U C 3 4 A HIERARCHY OF DEFINITIONS 73 df tp a A B C C A B u C F Kuratowski s ordered pair definition Definition 9 1 of Quine p 58 For proper classes it is not meaningful but is well defined for convenience Note that lt stands for whereas lt stands for lt and similarly for gt df op a F A B A A B df op a l lt A B gt A A4 B Define the union of a class Definition 5 5 p 16 of Takeuti and Zaring df uni a FU A a2 dy reyAyEeA df uni a U A xIlE yCxe y ye A Define the intersection of a class Definition 7 35 p 44 of Takeuti and Zaring df int a A 2 Vy yeA gt rey df int a A xlA yC ye A gt xe y Define a transitive class This should not be confused with a transitive relation which is a different concept Definition from p 71 of Enderton extended to classes df tr a H TrAS JACA df tr a Tr A lt gt U A LA Define a notation for a general binary relation Definition 6 18 p 29 of Takeuti and Zaring generalized to arbitrary classes This definition is well defined although not very meaningful when classes A and or B are proper The lack of parentheses or any other connective is not ambiguous since we are defining an atomic wff df br a k ARBO A B ER df br a ARB lt gt lt A B gt ec R Define an abstraction class of ordered pairs A special
238. oo inefficient to implement practically on a computer A practical algorithm for proving a smaller subset of geometry theorems those not involving concepts of order or continuity was discovered by Chinese mathematician Wu Wen tsiin in 1977 IO 26 CHAPTER 1 INTRODUCTION maticians know that they will never be replaced by computers providing one believes Roger Penrose s argument that a computer can never replace the brain 45 In fact elementary geometry is often considered a dead field for the simple reason that it is decidable On the other hand the undecidability of a theory does not mean that one cannot use a computer to search for proofs providing one is willing to give up if a proof is not found after a reasonable amount of time The field of automated theorem proving specializes in pursuing such computer searches Among the more successful results to date are those based on an algorithm known as Robinson s resolution principle 49 Automated theorem provers can be excellent tools for those willing to learn how to use them But they are not widely used in mainstream pure mathematics even though they could probably be useful in many areas There are several reasons for this Probably most important the main goal in pure mathematics is to arrive at results that are considered to be deep or important proving them is essential but secondary Usually an automated theorem prover cannot assist in this main goal and
239. oofs 2 3 A Trial Run Now you are ready to try out the Metamath program On all computer systems Metamath has a standard command line in terface CLI that allows you to interact with it You supply commands to the CLI by typing them on the keyboard and pressing your keyboard s Tf you ve looked at the set mm database you may have noticed another notation used for proofs The other notation is called compressed It reduces the amount of space needed to store a proof in the database and is described in Appendix B In the example above we use normal notation 2 3 A TRIAL RUN 41 return key after each line you enter The CLI is designed to be easy to use and has built in help features The first thing you should do is to use a text editor to create a file called demo0 mm and type into it the Metamath source shown on p Actually this file is included with your Metamath software package so check that first If you type it in make sure that you save it in the form of plain ASCH text with line breaks Most word processors will have this feature Next you must run the Metamath program Depending on your com puter system and how Metamath is installed this could range from clicking the mouse on the Metamath icon to typing run metamath to typing simply metamath Metamath s help invoke command describes alternate ways of invoking the Metamath program When you first enter Metamath it will be at the CLI wa
240. ooks a kind of necessary evil that was supposed to convince you of a certain key result Most of your homework consisted of exercises that gave you practice in the techniques and you were hardly ever asked to come up with a proof of your own You find yourself curious about advanced abstract mathematics You are driven by an inner conviction that it is important to understand and appreciate some of the most profound knowledge discovered by mankind But it seems very hard to learn something that only certain gifted longhairs can access and understand You are frustrated that it seems forever cut off from you Eventually your curiosity drives you to do something about it You set for yourself a goal of really understanding mathematics not just how to manipulate equations in algebra or calculus according to cookbook rules but rather to gain a deep understanding of where those rules come from In fact you re not thinking about this kind of ordinary mathematics at all but about a much more abstract ethereal realm of pure mathematics where famous results such as G del s incompleteness theorem and Cantor s different kinds of infinities reside 1 BO p vi 12 As quoted in 18 p 64 6 CHAPTER 1 INTRODUCTION You have probably read a number of popular books with titles like Infinity and the Mind 51 on topics such as these You found them inspiring but at the same time somewhat unsatisfactory They gave you a general idea
241. or copying and 150 CHAPTER 5 THE METAMATH PROGRAM pasting into another web page such as the tutorial pages on the Metamath web site Finally the command show statement brief_alt_html does the same as show statement brief_html for the alternate HTML symbol representation When two different types of pages need to be produced from a single database such as the Hilbert Space Explorer that extends the Metamath Proof Explorer extended variables may be declared in the t comment exthtmltitle HTML code exthtmlhome HTML code exthtmlbibliography filename When these are declared you also must declare exthtmllabel label that identifies the database statement where the extended section of the database starts in our example where the Hilbert Space Explorer starts During the generation of web pages for that starting statement and the state ments after it the HTML code assigned to exthtmltitle and exthtmlhome is used instead of that assigned to htmltitle and htmlhome respectively If you want to become familiar with these features you should study them in conjunction with the set mm database example in order to under stand the details that aren t precisely specified above such as exactly what the HTML code snippets should look like 5 8 2 write theorem list Command Syntax write theorem_list theorems_per_page number This command writes a list of all of the a and p statements in the dat
242. or this system can be found in 34 3 3 1 Propositional Calculus Axiom of Simplification ax 1 a F po w gt y Axiom of Distribution ax 2 a F p gt P gt x 7 Y7 4 gt 9 gt x Axiom of Contraposition ax 3 a F y gt 74 gt p gt gt 9 Rule of Modus Ponens maj e yw min e fy ax mp a Fw 3 3 2 Pure Predicate Calculus Axiom of Specialization ax 4 a F Vay y Axiom of Quantified Implication ax 5 a F Va Vayoy gt VapoVey Axiom of Quantified Negation ax 6 a F AVanVayy Axiom of Quantifier Commutation ax 7 a F VaVypoVyVay Rule of Generalization ax g 1 e Fy ax gen a F Vay 3 3 3 Equality and Substitution Axiom of Equality 1 ax 8 a F r y gt t z gt y 2z Axiom of Existence ax 9 a F Va r y gt Vary gt Axiom of Quantifier Substitution 3 3 THE AXIOMS IN THE METAMATH LANGUAGE 65 ax 10 a Va vw y gt VaypVypy Axiom of Variable Substitution ax 11 a F 7AVa w yo a y gt yoVa r y gt P Axiom of Quantifier Introduction 1 ax 12 a F AVz2 2 5 AV2 z y gt x y gt Yzzr y Axiom of Equality 2 ax 13 a H r y gt rEz gt yEz Axiom of Equality 3 ax 14 a y gt zer gt zey Axiom of Distinct Variables This axiom requires that two individual vari ables be distinct d xy ax 16 a Va r y gt pVzry Axiom of Quantifier Introduction 2 This axiom requires that the indi vidual variable not occur in the wff d x Y ax 17 a F poVay
243. ossibly using images of dogs and animals Intuitively the formu las seem to make sense But when you ask yourself What are the rules I need to get a computer to figure out whether this formula is correct you still don t know Certainly you don t ask the computer to imagine dogs and animals You look at some more advanced logic books Many of them have an introductory chapter summarizing set theory which turns out to be a pre requisite You need logic to understand set theory but it seems you also need set theory to understand logic These books jump right into prov ing rather advanced theorems about logic without offering the faintest clue about where the logic came from that allows them to prove these theorems Luckily you come across an elementary book of logic that halfway through after the usual truth tables and metaphors presents in a clear precise way what you ve been looking for all along the axioms They re di vided into propositional calculus also called sentential logic and predicate calculus also called first order logic with rules so simple and crystal clear that now you can finally program a computer to understand them Indeed they re no harder than learning how to play a game of chess As far as what you seem to need is concerned the whole book could have been written in five pages Now you think you ve found the ultimate source of mathematical truth So the axioms of mathematics consist of the
244. ot make any sense to do that This feature also means however that if you try to include a common file in several inner blocks the result might not be what you expect since only the first reference will be replaced with the included file unlike the include statement in most other computer languages Thus you would normally include common files only in the outermost block 4 4 4 Compressed Proof Format The proof notation presented in Section 4 3 is called a normal proof and in principle is sufficient to express any proof However proofs often con tain steps and subproofs that are identical This is particularly true in typical Metamath applications because Metamath requires that the math symbol sequence usually containing a formula at each step be separately constructed that is built up piece by piece As a result a lot of repetition often results The compressed proof format allows Metamath to take advantage of this redundancy to shorten proofs The specification for the compressed proof format is given in Appendix B Normally you need not concern yourself with the details of the com pressed proof format since the Metamath program will allow you to convert from the normal format to the compressed format with ease and will also au tomatically convert from the compressed format when proofs are displayed The overall structure of the compressed format is as follows 19On the Macintosh prior to Mac OS X a colon is used to sep
245. other to work out the details just make sure that you understand how you could do it in principle The answer is shown in the footnote on p If you actually do work it out you won t get exactly the same answer because we used a few simplifications such as discarding occurrences of double negation In the definitions below we have placed the asci Metamath source below each of the formulas to help you to become familiar with the notation in the database For simplicity the necessary f and d statements are not shown If you are in doubt use the show statement command in the Metamath program to see the full statement To understand the motivation for these definitions you should consult the references indicated Takeuti and Zaring 59 Quine 48 Bell and Ma chover 5 and Enderton I6 Our list of definitions is provided more for reference than as a learning aid However by looking at a few of them you can gain a feel for how the hierarchy is built up The definitions are a rep resentative sample of the 130 or so in set mm but they are complete with respect to the theorem examples we will present in Section 3 6 Also some are slightly different from but logically equivalent to the ones in set mm some of which have been revised over time to shorten them for example 3 4 1 Definitions for Propositional Calculus The symbols y Y and x represent wffs Our first definition introduces the biconditional connectivd also call
246. othesis to express 10Strictly speaking the d statement is also a hypothesis but it is never directly refer enced in a proof so we call it a restriction rather than a hypothesis to lessen confusion The checking for violations of d restrictions is automatic and built into Metamath s proof checking algorithm 106 CHAPTER 4 THE METAMATH LANGUAGE a logical truth such as assume x is prime that must be established in order for an assertion requiring it to also be true A variable must have its type specified in a f statement before it may be used in a e a or p statement There may be only one active f statement for a given variable Active is defined in Section 4 2 8 In ordinary mathematics theorems are often expressed in the form As sume P then Q where Q is a statement that you can derive if you start with statement PEJ In the Metamath language you would express math ematical statement P as a hypothesis a e Metamath language statement in this case and statement Q as a provable assertion a p statement Some examples of hypotheses you might encounter in logic and set theory are stmt1 f wff P stmt2 f set x stmt3 e P gt Q Informally these would be read Let P be a well formed formula Let x be an individual variable and Assume we have proved P gt Q The turnstile symbol F is commonly used in logic texts to mean a proof exists for To summar
247. parated by white space The quotes surround ing symbol match may be single or double quotes For example search b gt ch will list all statements whose labels begin with b and con tain the symbols gt and ch surrounding any symbol sequence including no symbol sequence The wildcards and are also available to match individual characters in labels and symbols respectively see help search in the Metamath program for details on their usage Optional command qualifiers all Also search e and f statements comments Search the comment that immediately precedes each label matched statement for symbol match In this case symbol match is an arbi trary non case sensitive character string Quotes around symbol match are optional if there is no ambiguity 5 5 Displaying and Verifying Proofs 5 5 1 show proof Command Syntax show proof label qualifiers see below This command displays the proof of the specified p statement in various formats The label may contain wildcard characters to match multiple statements Without any qualifiers only the logical steps will be shown i e syntax construction steps will be omitted in an indented format 5 5 DISPLAYING AND VERIFYING PROOFS 137 Most of the time you will use show proof label to see just the proof steps corresponding to logical inferences Optional command qualifiers essential The proof tree is trimmed of all f hypotheses before being displayed
248. phen Wolfram Mathematica A System for Doing Mathematics by Computer Addison Wesley Publishing Co Redwood City California second edition 1991 QA76 95 W65 1991 Larry Wos Ross Overbeek Ewing Lusk and Jim Boyle Automated Reasoning Introduction and Applications McGraw Hill Inc New York second edition 1992 QA76 9 A96 A93 1992 Index and auxiliary keywords 38 P3 97 119 96 98 Psi oa ma keyword 38 39 and auxiliary keywords 93 keyword 24 97 122 and keywords a statement 38 39 89 104 MOSTO SS A oO a g S EER E sE E E B oT f statemen 7 E E EE ZEEEE E p statement B as t commen v stateme inside comments 149 _ inside comments 121 inside comments in command lines inside proofs inside comments 119 althtmldef statement 149 156 althtmldir statement assign command 48 beep command close log command close tex command delete command erase command exit command exthtmlbibliography statement exthtmlhome statement T50 exthtmllabel statement 150 exthtmltitle statement 150 help command htmlbibliography statement 149 htmldef statement htmldir statement 149 htmlhome statement 149 htmltitle statement htmlvarcolors statement 149 improve command initialize eee latexdef statement 155 let command 143 mat
249. play before deletion Important Keep track of your work with a log file open log and save it frequently save new_proof write source because currently there is no undo command Hopefully there will be an undo command will be in a future version However you can use delete to reverse an assign and you can do delete floating_hypotheses then initialize all then unify all interactive to reinitialize bad unifications made accidentally or by bad assigns You cannot reverse a delete except by exit force then reentering the Proof Assistant to recover from the last save new_proof The following commands available in the Proof Assistant at the MM PA gt prompt to help you create your proof See the individual commands for more detail show new_proof all Displays the proof in progress You will use this command a lot see help show new_proof to become famil iar with its qualifiers The qualifiers unknown and not_unified are useful for seeing the work remaining to be done The combination 140 CHAPTER 5 THE METAMATH PROGRAM all unknown is useful identifying dummy variables that must be as signed or attempts to use illegal syntax when improve all is unable to complete the syntax constructions Unknown variables are shown as 1 2 assign step label Assigns an unknown step number with the state ment specified by label let variable variable symbol sequence Forces a symbol se quence to replace an unknown var
250. ple variable B9 Z0 09 LOT E3 in a formal system in ordinary mathematics 101 in predicate calculus in set theory Metamath variable declaration variable mae e ane a Bim ra variable type 99 variable type hypothesis 39 39 105 in a formal system 164 Venn diagram Wang Hao weak logic 17 H 2o a well formed ER DnA 20 B4 116 rea i well ordering Wen tsun Wu 25 white space af Whitehead ee rani STE Wiles Andrew 23 INDEX word processor 33 33 Wos Larry 26 Za 27 Zermelo Fraenkel set theory 9 BS ZFC set theory 195
251. plied rule explicit by stating only expressions of the above form are terms after defining term 38 CHAPTER 2 USING THE METAMATH PROGRAM tr f term r ts f term s wp f wff P wq f wff Q Define term and wff tze a term 0 tpl a term t r weq a wff t r wim a wff P gt Q State the axioms al a l t r gt t s gt r s a2 a l t 0 t Define the modus ponens inference rule f min e P maj e P gt Q mp a l Q Prove a theorem thi p l t t Here is its proof tt tze tpl tt weq tt tt weq tt a2 tt tze tpl tt weq tt tze tpl tt weq tt tt weq wim tt a2 tt tze tpl tt tt al mp mp A database is a set of one or more ASCII source files Here s a brief description of this Metamath database which consists of this single source file so that you can understand in general terms what is going on To understand the source file in detail you should read Chapter 4 The database is a sequence of tokens which are normally separated by spaces or carriage returns The only tokens that are built into the Metamath language are those beginning with These tokens are called keywords All other tokens are user defined and their names are arbitrary As you might have guessed the Metamath token starts a comment and ends a comment The Metamath tokens c v e f a and p specify statements th
252. plify the de velopment and to take advantage of the Metamath language In this section we will describe some common conventions used in set mm e The turnstile symbol meaning it is provable that is the first token of all assertions and hypotheses that aren t syntax constructions This is a standard convention in logic We mentioned this earlier but this symbol is bothersome to some people without a logic background It has no deeper meaning but just provides us with a way to distinguish syntax constructions from ordinary mathematical statements e A hypothesis of the form e K ypoVay should be read assume variable x is effectively not free in wf y Literally this says assume it is provable that y Vz vy This device lets us to avoid the complexities associated with the standard treat ment of free and bound variables The footnote on p discusses this further 3 5 TRICKS OF THE TRADE TT e A statement of one of the forms a F AVauv y p AVaer y should be read if x and y are distinct variables then This an tecedent provides us with a technical device to avoid the need for the d statement early in our development of predicate calculus permit ting symbol manipulations to be as conceptually simple as those in propositional calculus However the d statement eventually becomes a requirement and after that this device is rarely used e The statement d xy should be rea
253. point of view of our expository language The equivalent system of propositional calculus described in also uses the symbols Y etc to denote wff metavariables but in unlike here those are metavariables of the expository language and not primitive symbols of the formal system The first two statements define wffs if y and w are wits so is p w if y is a wff so is y The next three are the axioms of propositional calculus if p and w are wffs then F y gt y gt y is an axiomatic theorem etc The last is the rule of modus ponens if y and w are wffs and F y gt 4 and F y are theorems then w is a theorem The correspondence to ordinary propositional calculus is as follows We consider only provable statements of the form 9 T A with T defined as above The first term of the assertion A of any such statement is either wif or A statement for which first term is wf is a wff of proposi tional calculus and one where the first term is is a theorem scheme of propositional calculus The universe of this formal system also contains many other provable statements Those with distinct variable restrictions are irrelevant because propositional calculus has no constraints on substitutions Those that have logical hypotheses we call inferences when the logical hypotheses are of the form F w where w is a wff with the leading constant term wif removed Inferences other than the modus ponens r
254. positional calculus sometimes called sentential logic and predicate calculus sometimes called first order logic or quantifier theory Propositional calculus is a prerequisite for predicate calculus and predicate calculus is a prerequisite for set theory The version of set theory most commonly used is Zermelo Fraenkel set theory Here in a nutshell is what the axioms are all about in an informal way The connection between this description and symbols we will show you won t be immediately apparent and in principle needn t ever be Our description just tries to summarizes what mathematicians think about when they work with the axioms Logic is more or less the translation of what we would consider common sense into a rigorous set of axioms Suppose y Y and x the Greek letters phi psi and chi represent statements that are either true or false and x is a variable ranging over some group of mathematical objects sets integers real numbers etc In mathematics a statement really means a formula and w could be for example x 2 Logic makes assertions such as if y implies and w implies x then y implies x propositional calculus and if is true for all x then y is true for some x predicate calculus Set theory has to do with the manipulation of objects and collections of objects specifically the abstract imaginary objects that mathematics deals with such as numbers Everything that is claimed to exist
255. pressed proof can always be converted to a normal one and vice versa and the Metamath show proof commands work equally well with compressed proofs The compressed proof format is described in Appendix B MM PA gt show new_proof compressed Saas saS Clip out the proof below this line tze tpl weq a2 wim a1 mp ABCZADZAADZAEZJJKFLIA AGHH SSSSasss The proof of thi to clip out ends above this line Now we will exit the Proof Assistant Since we made changes to the proof it will warn us that we have not saved it In this case we don t care MM PA gt exit Warning You have not saved changes to the proof Do you want to EXIT anyway Y N lt N gt y Exiting the Proof Assistant Type EXIT again to exit Metamath The Proof Assistant has several other commands that can help you while creating proofs See Section 5 6 for a list of them A command that is often useful is minimize_with brief which tries to shorten the proof It can make the process more efficient by letting you write a somewhat sloppy proof then clean up some of the fine details of op timization for you although it can t perform miracles such as restructuring the overall proof 2 5 A NOTE ABOUT EDITING A DATABASE FILE 53 2 5 A Note About Editing a Database File Once your source file contains proofs there are some restrictions on how you can edit it so that the proofs remain valid Pay particular attention to these rules since otherwis
256. preted as 70 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED ya y or yzy 7 Other texts often use y y to indicate our y z y but this notation is even more ambiguous since there is no explicit indication of what is being substituted df sb a F y r po t y gt y Ada a yAy df sb a l Ly x ph lt gt x y gt ph AE x x y ph Define existential uniqueness there exists exactly one Note that y is a variable distinct from x and not occurring in y df eu a F dlapodyVa por y df eu a E x ph lt gt E y A x ph lt gt x y 3 4 3 Definitions for Set Theory The symbols x y z and w represent individual variables of predicate cal culus which in set theory are understood to be sets To make set theory more practical we introduce the notion of a class A class is either a set variable such as x or an expression of the form x y called an abstrac tion class Note that sets i e individual variables always exist this is a theorem of logic namely Jyy x for any set x whereas classes may or may not exist i e Jyy A may or may not be true If a class does not exist it is called a proper class Definitions df clab df cleq and df clel can be used to convert an expression containing classes into one containing only set variables and wff metavariables The symbols A B C D F G and R are metavariables that range over classes A class metavariabl
257. proof is in whether a computer program works or not With 14 CHAPTER 1 INTRODUCTION easily accessible computer based abstract mathematics a mathematician could say to a crank don t bother me until you ve demonstrated your claim on the computer 1 1 3 An Impossible Dream Even quite basic theorems would demand almost unbelievably vast books to display their proofs ROBERT E Epwarp Oh of course no one ever really does it It would take forever You just show that you could do it that s sufficient THE IDEAL MATHEMATICIAN There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 2000 3000 The formula would be forbiddingly long even if one knows the def initions and is asked to simplify the long formula according to them chances are he will make errors and arrive at some incor rect result HAo Wand The Principia Mathematica was the crowning achievement of the formalists It was also the deathblow of the formalist view Russell failed in three enormous volumes to get beyond the elementary facts of arithmetic He showed what can be done in principle and what cannot be done in practice If the mathemati cal process were really one of strict logical progression we would still be counting our fingers One theoretician estimates for instance that a demonstration of one of Ramanujan s conjectures assuming set theory and ele menta
258. pt that we don t specify a statement instead the proof we re working on is displayed MM PA gt assign 1 mp To undo the assignment DELETE STEP 5 and INITIALIZE UNIFY 48 CHAPTER 2 USING THE METAMATH PROGRAM if needed 3 min 2 4 maj 2 gt t t The assign command above means assign step 1 with the statement whose label is mp Note that step renumbering will constantly occur as you assign steps in the middle of a proof in general all steps from the step you assign until the end of the proof will get moved up In this case what used to be step 1 is now step 5 because the partial proof now has five steps the four hypotheses of the mp statement and the mp statement itself Let s look at all the steps in our partial proof MM PA gt show new_proof lemmon all 1 wit 2 2 wif t t 3 7 2 Tr l 2 gt t t 5 1 2 3 4 mp a l t t The symbol 2 is a temporary variable that represents a symbol sequence not yet known In the final proof all temporary variables will be eliminated The general format for a temporary variable is followed by an integer Note that is not a legal character in a math symbol see Section 4 2 1 p 97 so there will never be a naming conflict between real symbols and temporary variables Unknown steps 1 and 2 are constructions of the two wffs used by the modus ponens rule As you will see at the end of this section the Proof Assistant can usually figure
259. r s operating system if it has a command line interface For example on a VAX VMS system MM gt dir will print disk directory contents Note that this feature will not work on the Macintosh prior to Mac OS X which does not have a command line interface For your convenience the trailing quote is optional 134 CHAPTER 5 THE METAMATH PROGRAM 5 2 13 Size Limitations in Metamath In general there are no fixed predefined limits on how many labels tokens statements etc that you may have in a database file The Metamath pro gram uses 32 bit variables 64 bit on 64 bit CPUs as indices for almost all internal arrays which are allocated dynamically as needed 5 3 Reading and Writing Files The following commands create new files the open commands the write commands the html alt_html brief_html1 brief_alt_html quali fiers of show statement and midi The following commands append to files previously opened the tex qualifier of show proof and show new_proof the tex and simple_tex qualifiers of show statement the close com mands and all screen dialog between open log and close log The commands that create new files will not overwrite an existing file name but will rename the existing one to filename 1 An existing filename 1 is renamed filename 2 etc up to filename 9 An existing filename 9 is deleted This makes recovery from mistakes easier but also will clutter up your directory so occasionally you may wa
260. re s no ambiguity MM gt search conjunction comments 634 wa a wff definition to include conjunction and 636 df an a Define conjunction and 654 iman p Express implication in terms of conjunction 655 annim p Express conjunction in terms of implication 687 pm3 2 p antecedents with conjunction Theorem 751 anor p Conjunction in terms of disjunction de Morg etc Now we will start to look at some details Let s look at the first axiom of propositional calculus MM gt show statement ax 1 full MM gt sh st ax 1 full 3 8 EXPLORING THE SET THEORY DATABASE 85 Statement 19 is located on line 881 of the file set mm Axiom _Simp_ Axiom A1 of Margaris p 49 One of the 3 axioms of propositional calculus The 3 axioms are also 19 ax 1 a ph gt ps gt ph Its mandatory hypotheses in RPN order are wph f wff ph wps f wff ps The statement and its hypotheses require the variables ph ps The variables it contains are ph ps Compare this to ax 1 on p You can see that the symbol ph is the ASCII notation for y etc To see the mathematical symbols for any expression you may typeset it in ATEX type help tex for instructions or easier just use a text editor to look at the comments where symbols are first introduced in set mm The hypotheses wph and wps required by ax 1 mean that variables ph and ps must be wffs Next we ll pick a simple theorem of
261. re properly called an inference rather than a theorem 4 2 THE BASIC KEYWORDS 107 label a constant math symbol math symbol label p constant math symbol math symbol proof An assertion always requires a label The expression in an assertion consists of a constant followed by a sequence of zero or more math symbols Each math symbol including constant must be a previously declared constant or variable In addition each math symbol must be active which will be covered when we discuss scoping statements in Section 4 2 8 A a statement is usually a definition of syntax for example if P and Q are wffs then so is P Q an axiom of ordinary mathematics for example x x or a definition of ordinary mathematics for example x y means g y A p statement is a claim that a certain combination of math symbols follows from previous assertions and is accompanied by a proof that demonstrates it Assertions can also be referenced in later proofs in order to derive new assertions from them The label of an assertion is used to refer to it in a proof Section 4 3 will describe the proof in detail Assertions also provide the primary means for communicating the math ematical results in the database to people Proofs when conveniently dis played communicate to people how the results were arrived at The a Statement Axiomatic assertions a statements represent the starting points from which other asser
262. realiza tion that there is another whole field of study that you need just to get to the point at which you can understand the axioms of set theory The field is called logic In fact some of the books did recommend it as a prerequisite but it just didn t sink in You assumed logic was well just logic something that a person with common sense intuitively understood Why waste your time reading boring treatises on symbolic logic the manipulation of 1 s and 0 s that computers do when you already know that But this is a different kind of logic quite alien to you The subject of NAND and NOR gates is not even touched upon or in any case has to do with only a very small part of this field So your quest continues Skimming through the first couple of introduc tory books you get a general idea of what logic is about and what quantifiers for all there exists mean but you find their examples somewhat triv ial and mildly annoying all dogs are animals some animals are dogs and such But all you want to know is what the rules are for manipulating the symbols so you can apply them to set theory Some formulas describing 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 11 the relationships among quantifiers 4 and V are listed in tables along with some verbal reasoning to justify them Presumably if you want to find out if a formula is correct you go through this same kind of mental reasoning process p
263. recise steps that leave little doubt that the results follow from the axioms You might think that this would be the finest breakdown we can achieve in mathematics However there is more to the proof than meets the 2 2 YOUR FIRST FORMAL SYSTEM 37 eye Although our axioms were rather simple a lot of verbiage was needed before we could even state them we needed to define term wff and so on In addition there are a number of implied rules that we haven t even mentioned For example how do we know that step 3 of our proof follows from axiom Al There is some hidden reasoning involved in determining this Axiom Al has two occurrences of the letter t One of the implied rules states that whatever we substitute for t must be a legal term The expression t 0 is pretty obviously a legal term whenever t is but suppose we wanted to substitute a huge term with thousands of symbols Certainly a lot of work would be involved in determining that it really is a term but in ordinary formal proofs all of this work would be considered a single step To express our axiom system in the Metamath language we must de scribe this auxiliary information in addition to the axioms themselves Meta math does not know what a term or a wff is In Metamath the specifi cation of the ways in which we can combine symbols to obtain terms and wffs are like little axioms in themselves These auxiliary axioms are expressed in the same notatio
264. rm to use in place of t and substitute it for t throughout the proof 36 CHAPTER 2 USING THE METAMATH PROGRAM Let s discuss what we have done here The axioms of our theory Al and A2 are trivial and obvious Everyone knows that adding zero to something doesn t change it and also that if two things are equal to a third then they are equal to each other In fact stating the trivial and obvious is a goal to strive for in any axiomatic system From trivial and obvious truths that everyone agrees upon we can prove results that are not so obvious yet have absolute faith in them If we trust the axioms and the rules we must by definition trust the consequences of those axioms and rules if logic is to mean anything at all Our rule of inference modus ponens is also pretty obvious once you understand what it means If we prove a fact P and we also prove that P implies Q then Q necessarily follows as a new fact The rule provides us with a means for obtaining new facts i e theorems from old ones The theorem that we have proved t t is so fundamental that you may wonder why it isn t one of the axioms In some axiom systems of arithmetic it 7s an axiom The choice of axioms in a theory is to some extent arbitrary and even an art form constrained only by the requirement that any two equivalent axiom systems be able to derive each other as theorems We could imagine that the inventor of our axiom system originally included t t as
265. rs The set of keyword tokens is c v f e d a p C and The last four are called auxiliary The current version of the Metamath program 0 07 30 implements an older speci fication with a slightly more general syntax than described here Among the differences missing white space around keyword tokens is sometimes tolerated and a variable may be used before its type is specified by a f statement this feature should never be used in practice A future version of the program is expected to conform to the current specification 3 As quoted in 14 p 273 4 1 SPECIFICATION OF THE METAMATH LANGUAGE 93 or preprocessing keywords A label token consists of any combination of letters digits and the characters hyphen underscore and period A math symbol token may consist of any combination of the 93 printable standard ASCII characters other than All tokens are case sensitive 4 1 2 Preprocessing begins a comment and ends a comment Comments may contain any of the 94 printable characters and white space except they may not contain the 2 character sequences or Comments are ignored treated like white space for the purpose of parsing Two special characters inside comments and control the typesetting of comments and are discussed on p 119 They may be ignored for the purpose of parsing A file inclusion command consists of followed by a file name fol lowed by
266. ry analysis would take about two thousand pages the length of a deduction from first principles is nearly inconceivable The probabilists argue that any very long proof can at best be viewed as only probably correct RICHARD DE MILLO ET ALE 16 I5 p 68 17 T3 p 40 18 63 p 140 19 74 pp 269 271 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 15 A number of writers have conveyed the impression that the kind of ab solute rigor provided by Metamath is an impossible dream suggesting that a complete formal verification of a typical theorem would take millions of steps in untold volumes of books Even if it could be done the thinking sometimes goes all meaning would be lost in such a monstrous tedious verification These writers assume however that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms This is not necessary There is no reason not to make use of previously proved theorems rather than proving them over and over Just as important definitions can be introduced along the way allow ing very complex formulas to be represented with few symbols Not do ing this can lead to absurdly long formulas For example G del s incom pleteness theorem which can be expressed with a small number of de fined symbols would require about 20 000 primitive symbols to express ite An extreme example
267. s o ooa aaa 1 3 2 Automated Theorem Provers 1 3 3 Proof Verifiers o aooaa a 1 4 Mathematics and Metamath a a aoa a a a 1 4 1 Standard Mathematics 1 4 3 Metamath and Its Philosoph oa S 2 Using the Metamath Program 2 1 Installation 2 6 saaa cocoes aea ea a a e e a 2 2 Your First Formal System aa From Nothing to Zero 2 2 2 Converting It to Metamath Z 2 3 1 Some Hints for Using the Command Line Interface 2 4 Your First Proof saast sees es sa ee 2 5 A Note About Editing a Database File aoaaa iii 3__Abstract Mathematics Revealed 3 1 Logic and Set Theory 3 2 The Axioms for All of Mathematics 3 2 1 Propositional Calculus 3 2 2 Predicate Calculus 3 2 3 Equality 2 4 Set Theory 3 3 The Axioms in the Metamath Language w 3 3 1 Propositional Calculus 3 3 2 Pure Predicate Calculus 3 3 3 Equality and Substitution 3 4 Set Theory 3 3 5 That s It se 3 4 A Hierarchy of Definitions 3 4 1 Definitions for Propositional Calculus 3 4 2 Definitions for Predicate Calculus 3 4 3 Definitions for Set Theory 3 5 Tricks of the Trade 3 6 A Theorem Sampler 3 7 Axioms for Real and Complex Numbers 3 8 Exploring the Set Theory Database 3 8 1 A Note on Compact Proof
268. s There are a number of ways that Metamath can be used with standard mathematics The most satisfying way philosophically is the start at the very beginning and develop the desired mathematics from the axioms of logic and set theory This is the approach taken in the set mm module provided with the Metamath software Among other things this module builds up to the axioms of real and complex numbers see Section 3 7 and a standard development of analysis for example could start at that point using it as a basis Besides this philosophical advantage there are practical advantages to having all of the tools of set theory available in the supporting infrastructure On the other hand you may wish to start with the standard axioms of a mathematical theory without going through the set theoretical proofs of those axioms You will need mathematical logic to make inferences but if you wish you can simply introduce theorems of logic as axioms wherever you need them with the implicit assumption that in principle they can be proved if they are obvious to you If you choose this approach you will probably want to review the notation used in set mm so that your own notation will be consistent with it 1 4 2 Other Formal Systems Unlike some programs Metamath is not limited to any specific area of math ematics nor committed to any particular mathematical philosophy such as classical logic versus intuitionism nor limited say to expressions
269. s f step 20 The parent assertion of the target hypothesis is ax mp a step 22 The source assertion before substitution was ax 1 a ph gt ps gt ph The following substitutions were made to the source assertion Variable Substituted with ph ph ps ph gt ph 3 8 EXPLORING THE SET THEORY DATABASE 87 The target hypothesis before substitution was min e ph The following substitution was made to the target hypothesis Variable Substituted with ph ph gt ph gt ph gt ph This shows the substitutions made to the variables in ax 1 References are made to steps 19 and 20 which are not shown in our proof display To see these steps you can display the proof without the essential qualifier Let s look at a slightly more advanced proof of propositional calculus Note that is the symbol for A logical AND also called conjunction MM gt show statement prth full Statement 1521 is located on line 4730 of the file set mm Theorem 3 47 of Whitehead and Russell called praeclarum theorema by Leibniz prth p l ph gt ps ch gt th gt ph ch gt ps th Its mandatory hypotheses in RPN order are wph f wff ph wps f wff ps wch f wff ch wth f wff th Its optional hypotheses are wet The statement and its hypotheses require the variables ph ps ch th These additional variables are allowed in its proof et The vari
270. s derived as a theorem from earlier statements we omit its presumed proof Variable R may be used in its proof if desired although this would probably have no advantage in propositional calculus Note that the sequence of mandatory hypotheses in RPN order is still wp wq maj min i e wr is omitted and this sequence is still assumed whenever assertion mp is referenced in a subsequent proof wp f wff P wq f wff Q wr f wff R maj e l P gt Q min e P mp p l Q Every frame is an extended frame but not every extended frame is a frame as this example shows The underlying frame for an extended frame is obtained by simply removing all statements containing optional variables Any proof referencing an assertion will ignore any extensions to its frame which means we may add or delete optional hypotheses at will without affecting subsequent proofs The conceptually simplest way of organizing a Metamath database is as a sequence of extended frames The scoping statements and can be used to delimit the start and end of a frame leading to the following possible structure for a database v and c statements extended frame extended frame In practice this structure is inconvenient because we have to repeat any f e and d statements over and over again rather than stating them once for use by several assertions The scoping statements which we will discuss next allow th
271. s their needs and there may be a general feeling that such software could not even exist On the one hand there are the practical computer algebra systems which can perform amazing symbolic manipulations in algebra and calculus yet can t prove the simplest existence theorem if the idea of a proof is present at all On the other hand there are specialized automated theorem provers that technically speaking may generate correct proofs But sometimes their specialized input notation may be cryptic and their output perceived to be long inelegant incomprehensible proofs The output may be viewed with suspicion since the program that generates it tends to be very large and its size increases the potential for bugs Such a proof may be considered trustworthy only if independently verified and understood by a human but no one wants to waste their time on such a boring unrewarding chore 1 2 1 Trusting the Computer I continue to find the quasi empirical interpretation of com puter proofs to be the more plausible Since not everything that claims to be a computer proof can be accepted as valid what are the mathematical criteria for acceptable computer proofs THOMAS TymoczKkd In some cases computers have been essential tools for proving famous theorems But if a proof is so long and obscure that it can be verified in a practical way only with a computer it is vaguely felt to be suspicious For example proving the famous
272. se axioms of logic together with the axioms of ZF set theory By now you ve also been able to figure out how to translate the ZF axioms from English into the actual symbols of logic which you can now manipulate according to precise easy to understand rules Of course you still don t understand how all of mathematics can be derived from set theory but maybe this will reveal itself in due course You eagerly set out to program the axioms and rules into a computer and start to look at the theorems you will have to prove as the logic is developed All sorts of important theorems start popping up the deduction theorem the substitution theorem the completeness theorem of propositional calcu lus the completeness theorem of predicate calculus Uh oh there seems to be trouble They all get harder and harder and not one of them can be derived with the axioms and rules of logic you ve just been handed Instead they all require metalogic for their proofs a kind of mixture of logic and set theory that allows you to prove things about the axioms and theorems of logic rather than with them You plow ahead anyway A month later you ve spent much of your free time getting the computer to verify proofs in propositional calculus You ve programmed in the axioms but you ve also had to program in the 12 CHAPTER 1 INTRODUCTION deduction theorem the substitution theorem and the completeness theorem of propositional calculus w
273. self By eliminating the definition we don t mean repeatedly applying it to specific m and n but rather showing the explicit closed form set theoretical expression that m n represents that will work for any m and n and that does not have a sign on its right hand side For 20Two nice systems that make the transition from intuitionistic and other weak frag ments to classical logic just by adding axioms are given in 50 4 5 APPENDIX AXIOMS VS DEFINITIONS 127 a recursive definition like this not to be circular creative there are some hidden underlying assumptions we must make for example that the natu ral numbers have a certain kind of order In set mm we chose to start with the direct though complex and nonintuitive definition then derive from it the standard recursive definition The end result is the same but we completely eliminate the rather complex metalogic that justifies the recur sive definition For a mathematician recursive definitions are more efficient and intuitive than direct ones once the metalogic has been learned or pos sibly just accepted as correct However it was felt that direct definition in set mm maximizes rigor by minimizing metalogic It can be eliminated effortlessly something difficult to do with a recursive definition 21The closed form definition used in set mm for the addition operation on ordinals of which natural numbers are a subset is df oadd a F 2 y z e e OnA y
274. sfied and are checked by the proof verifier when expressions are substituted for them in a proof and the d statements them selves are never referenced directly in a proof A frame with a p provable statement requires a proof as part of the p statement Sometimes in a proof we want to make use of temporary or dummy variables that do not occur in the p statement or its mandatory hypotheses To accommodate this we define an extended frame as a frame together with zero or more d and f statements that reference variables not among the mandatory variables of the frame Any new variables referenced are called the optional variables of the extended frame If a f statement references an optional variable it is called an optional hypothesis and if one or both of the variables in a d statement are optional variables it is called an optional disjoint variable restriction Properties 2 3 and 4 for a frame also apply to an extended frame The concept of optional variables is not meaningful for frames with a statements since those statements have no proofs that might make use of them There is no restriction on including optional hypotheses in the ex tended frame for a a statement but they serve no purpose 4 2 THE BASIC KEYWORDS 111 The following set of statements is an example of an extended frame which contains an optional variable R and an optional hypothesis wr In this example we suppose the rule of modus ponens is not an axiom but i
275. shortest possible 3 2 THE AXIOMS FOR ALL OF MATHEMATICS 59 axiom schemes Each Greek letter in the axioms may be substituted with a more complex wff to result in another axiom For example substituting y x for in the first axiom yields y gt x gt y gt 7A y gt x which is still an axiom To deduce new true statements theorems from the axioms a rule called modus ponens is used This rule states that if the wff y is an axiom or a theorem and the wff y w is an axiom or a theorem then the wff w is also a theorem As a non mathematical example of modus ponens suppose we have proved or taken as an axiom Bob is a man and separately have proved or taken as an axiom If Bob is a man then Bob is a human Using the rule of modus ponens we can logically deduce Bob is a human From Metamath s point of view the axioms and the rule of modus po nens just define a mechanical means for deducing new true statements from existing true statements and that is the complete content of propositional calculus as far as Metamath is concerned You can read a logic textbook to gain a better understanding of their meaning or you can just let their meaning slowly become apparent to you after you use them for a while It is actually rather easy to check to see if a formula is a theorem of propositional calculus Theorems of propositional calculus are also called tautologies The technique t
276. simply state that R is a complete ordered subfield of C leading to redundant axioms when this phrase is completely expanded out In fact I have not seen a text with the axioms in the explicit form above It is possible that one or more 3 8 EXPLORING THE SET THEORY DATABASE 83 of the axioms above are redundant or could be made weaker if you discover an improvement please let me know and I will properly acknowledge your contribution Update Feb 2005 The third axiom 0 is a real number is redundant see the http metamath org web site for details 3 8 Exploring the Set Theory Database At this point you may wish to study the set mm file in more detail Pay particular attention to the assumptions needed to define wffs which are not included above the variable types f statements and the definitions that are introduced Start with some simple theorems in propositional calculus making sure you understand in detail each step of a proof Once you get past the first few proofs and become familiar with the Metamath language any part of the set mm database will be as easy to follow step by step as any other part you won t have to undergo a quantum leap in mathematical sophistication to be able to follow a deep proof in set theory Next you may want to explore how concepts such as natural numbers are defined and described This is probably best done in conjunction with standard set theory textbooks which can h
277. stant only The Metamath language allows variables to be substituted with empty symbol sequences However in many formal systems this will never hap pen in a valid proof Allowing for this possibility increases the likelihood of ambiguous unifications during proof creation and you may want to set empty_substitution off to help make the process more efficient With this mode set you may not be able to create some proofs in formal systems that allow empty substitutions An example would be a system that imple ments a Deduction Rule and in which deductions from empty assumption lists would be permissible The MIU system described in Appendix D is another example Note that this command does not affect the way proofs are verified with the verify proof command Outside of the Proof Assistant substitution of empty sequences for math symbols is always allowed 5 6 4 set search limit Command Syntax set search_limit number This command is available outside the Proof Assistant but affects the Proof Assistant only This command sets a parameter that determines when the improve com mand in Proof Assistant mode gives up If you want improve to search harder you may increase it The show settings command tells you its current value 5 6 5 show new_proof Command Syntax show new_proof qualifiers see below This command available only in Proof Assistant mode displays the proof in progress It is identical to the show proof command e
278. sult of A R D Mathias on the system of Bourbaki and the text has been updated accordingly p 15h Bob also pointed out a clarification of the literature regarding category theory and inaccessible cardinals p 31 and a misleading statement was removed from the text Specifically contrary to a statement in previous editions it is possible to express There is a proper class of inaccessible cardinals in the language of ZFC This can be done as follows For every set x there is an inaccessible cardinal such that is not in x Bob writes This axiom is how Grothendieck presents category theory To each inaccessible cardinal k one associates a Grothendieck universe U U consists of those sets which lie in a tran sitive set of cardinality less than Instead of the category of all groups one works relative to a universe considering the category of groups of cardinality less than Now the category whose objects are all categories relative to the universe U kK will be a category not relative to this universe but to the next universe All of the things category theorists like to do can be done in this framework The only controversial point is whether the Grothendieck axiom is too strong for the needs of category theo rists Mac Lane argues that one universe is enough and Fefer man has argued that one can get by with ordinary ZFC I don t 3Private communication Nov 30 2002 PRE
279. t 344 Meredith C A metalanguage metalogic 64 126 metalogical completeness 17 1 Metamath vii xi 2 j r EBE m 107 L20 f123 125 as a formal system bugs commands installation limitations of version 0 07 30 CERS ERE EEE BERS EE 2 ase Ti Be E 0 02 LT E3 TAT ES memory limits 134 memory usage representation of numbers self description specification using as a math editor 120 metamathematics metatheorem 12 metavariable 34 Millay Edna MIU system Miyaoka Yoichi Mizar xiii xiii Ey modal logic 17 model Reig modus ponens Monaco font Monk J Donald 83 monospaced font Munkres James R 3 natural number INDEX negation Nemesszeghy E Z nested block nesting level non scoping statement 112 non trivial theory 100 normal proof null set number theory object 125 object EAR omega w 57 66 74 80 one to one function onto function operating system command operation operator precedence optional disjoint variable restriction 110 optional hypothesis optional variable ordered pair ordinal addition ordinal number ordinal predicate OTTER outermost block pair parsing Metamath Pasch s axiom Pavici M Peano s postulates Penrose Roger Perelman Grigory Peterson Jeremy George Pierce s axiom 120 pl
280. t YyYz v C7 T F np gt Yre p C8 T F x y gt xz z gt y 2 C9 T F Yx x y gt War z gt y z gt Yzy 2 C10 2 T F Vala y gt Yz p gt 9 C11 8 T F Vrz y gt Vay gt Yy p C12 T F x y gt Raz gt Ryz C13 T F x y gt Rzx gt Rzy C15 T F Cee y gt x y gt Y gt Yz x y gt C16 Hiz y T F War y gt p gt Yz p C5 x p T F p gt Ve y MP 2 7 F 2 gt E p E p Gen T F y F Ve y While it is known that these axioms are metalogically complete it is not known whether they are independent i e none is redundant in the metalogical sense specifically whether any axiom possibly with additional non mandatory distinct variable restrictions for use with any dummy vari ables in its proof is provable from the others Note that metalogical inde pendence is a stronger requirement than independence in the usual logical sense Not all of the above axioms are logically independent for example C9 can be proved as a metatheorem from the others outside the formal system by combining the possible cases of distinct variables C 3 6 Example 4 Adding Definitions There are several ways to add definitions to a formal system Probably the most proper way is to consider definitions not as part of the formal s
281. t are the longest this command ferrets out one of them and shows it to you There may be a sense in which the longest path length is related to how deep theorem is Finally we might be curious about what proofs depend the theorem prth If it is never used later on we could eliminate it as redundant if it has no intrinsic interest by itself MM gt show usage prth Statement prth is directly referenced in the proofs of 4 statements anim12d mo 2mo ssxp tfrlem5 climunii climadd Thus prth is used by 7 proofs and indirectly by many more that make use of those proofs and so on The recursive qualifier gives you all of them 3 8 1 A Note on Compact Proof Format The present version of Metamath 0 07 30 will display proofs in a compact format whenever the proof is stored in compressed format in the database It may be be slightly confusing unless you know how to interpret it A future version may eliminate this format from displays For example if you display the complete proof of theorem id1 it will start off as follows 90 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED MM gt show proof id1 lemmon all 1 wph f wff ph 2 wph f wff ph 3 wph f wff ph 4 2 3 wi 1 a wff ph gt ph 5 1 4 wi 2 a wff ph gt ph gt ph 6 1 a wff ph gt ph etc Step 4 has a local label 1 assigned to it Later on at step 6 the label 1 is referenced instead of displaying the explicit proof for tha
282. t for a theorem then lt A NAME Monk gt must be present in the file if not a warning message is given Single or double quotes surround the HTML code strings and the filename string Strings too long for a line may be broken up as descibed for the latexdef statement in Appendix A That Appendix also describes how to handle strings containing quote characters The t comment may also contain TEX definitions with latexdef statements see Appendix A that are ignored for HTML output Several other HTML related qualifiers exist for the show statement com mand The command show statement label alt_html does the same as show statement label html except that the HTML code for the symbols is taken from althtmldef statements instead of htmldef statements in the t comment althtmldef math token as HTML code althtmldef math token as HTML code This feature is useful when an alternate representation of symbols is desired for example one that uses Unicode entities instead of GIF images Associated with althtmldef are the statements htmldir directoryname althtmldir directoryname giving the directories of the GIF and Unicode versions respectively their purpose is to provide cross linking between the two versions in the generated web pages The command show statement brief_html invokes a special mode that just produces definition and theorem lists ac companied by their symbol strings in a format suitable f
283. t step This technique takes advantage of the fact that steps in a proof often repeat especially during the construction of wffs The compact format reduces the number of steps in the proof display and may actually be preferred by some people which is one reason that it is still displayed by the program If you want to see the normal format with the true step numbers you can use the following workaround MM gt save proof id1 normal The proof of idi has been reformatted and saved internally Remember to use WRITE SOURCE to save it permanently MM gt show proof id1 lemmon all 1 wph f wff ph 2 wph f wff ph 3 wph f wff ph 4 2 3 wi a wif ph gt ph 5 1 4 wi a wff ph gt ph gt ph 6 wph f wff ph 7 wph f wff ph 8 6 7 wi a wff ph gt ph etc Note that the original 6 steps are now 8 steps However the format is now the same as that described in Chapter 2 Chapter 4 The Metamath Language Thus mathematics may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true BERTRAND RussEuE Probably the most striking feature of the Metamath language is its al most complete absence of hard wired syntax Metamath does not under stand any mathematics or logic other than that needed to construct finite sequences of symbols according to a small set of simple built in rules The only rule it uses in a proof is the substitution of an e
284. t the reader user knows that all details are contained in its computer database available as needed it does not demand that the user know everything but conveniently makes available those por tions that are of interest As the body of all mathematical knowledge grows larger and larger no one individual can have a thorough grasp of its entirety Metamath can finalize and put to rest any questions about the validity of any part of it and can make any part of it accessible in principle to a non specialist A Personal Note Why did I develop Metamath I enjoy abstract mathematics but I some times get lost in a barrage of definitions and start to lose confidence that my proofs are correct Or I reach a point where I lose sight of how anything Pm doing relates to the axioms that a theory is based on and am sometimes suspicious that there may be some overlooked implicit axiom accidentally in troduced along the way as happened historically with Euclidean geometry whose omission of Pasch s axiom went unnoticed for 2000 years I3 p 160 Pm also somewhat lazy and wish to avoid the effort involved in re verifying the gaps in informal proofs left to the reader I prefer to figure them out just once and not have to go through the same frustration a year from now when I ve forgotten what I did Metamath provides better recovery of my efforts than scraps of paper that I can t decipher anymore But mostly I find very appealing the id
285. tep assign 2 label the antepenultimate and assign 0 label is the same as assign last label Optional command qualifier no_unify do not prompt user to select a unification if there is more than one possibility This is useful for noninteractive command files Later the user can unify all interactive The assignment will still be au tomatically unified if there is only one possibility and will be refused if unification is not possible 5 6 7 match Command Syntax match step step max_essential_hyp number and match all essential max_essential_hyp number This command available in the Proof Assistant only shows what state ments can be unified with the specified step s Note In its current form this command is not very useful because of the large number of matches it reports I rarely use it It may be enhanced in the future Optional command qualifiers max_essential_hyp number filters out of the list any statements with more than the specified number of e hypotheses essential_only in the match all statement only the steps that would be listed in show new_proof essential display are matched 5 6 8 let Command Syntax let variable variable symbol sequence and let step step symbol sequence 144 CHAPTER 5 THE METAMATH PROGRAM These commands available in the Proof Assistant only assign a tem porary variable or unknown step with a specific symbol sequence They are useful in the middle of creat
286. ters xz y z and w as is common practice in the literature To prevent confusion it might be best at this point to think of the vari ables of Metamath as metavariables because they are not quite the same as the variables we are introducing here A meta variable in Metamath can be a wff or an individual variable as well as many other things in gen eral it represents a kind of place holder for an unspecified sequence of math symbols In predicate calculus we extend the definition of a wff If y is a wff and x and y are variables then Vay x y and x y are wffs Note that these three new types of wffs can be considered starting wffs from which we can build other wffs with and The concept of a starting wff was absent in propositional calculus But starting wff or not all we are really concerned with is whether our wffs are correctly constructed according to these mechanical rules In standard texts of logic there are two axioms of predicate calculus Va p x gt p y where y is properly substituted for x Vae p gt Y gt p gt Yz y where x is not free in y Now at first glance this seems simple just two axioms However condi tional clauses are attached to each axiom describing requirements that may seem puzzling to you In addition the first axiom puts a variable symbol in parentheses after each wff seemingly violating our definition of a wff this is just an informal way of referring
287. th the state ment that has the same subscript Each pair of matched scoping statements defines a region of the database called a block Blocks can be nested inside of other blocks in the example the block defined by 4 and 4 is nested inside the block defined by 3 and 3 as well as inside the block defined by and In general a block may be empty it may contain only non scoping statements or it may contain any mixture of other blocks and non scoping statements This is called a recursive definition of a block Associated with each block is a number called its nesting level that indicates how deeply the block is nested The nesting levels of the blocks in our example are as follows lt lt The entire database is considered to be one big block the outermost block with a nesting level of 0 The outermost block is not bracketed by scoping statements 12 Those statements other than and 13 The language was designed this way so that several source files can be joined together more easily 4 2 THE BASIC KEYWORDS 113 All non scoping Metamath statements become recognized or active at the place where they appear 4 Certain of these statement types become inactive at the end of the block in which they appear these statement types are c v d e and f The other statement types remain active forever i e through the end of the database they are a and p
288. that establishes it If the proof of a statement is questionable or vague we can t have absolute confidence in the truth that the statement claims Let us look at some traditional ways of expressing proofs not been otherwise shown to be inconsistent the axiom of choice turns out to be false p 23 The show trace_back command of the Metamath program allows you to find out whether the axiom of choice or any other axiom was assumed by a proof 2 27 p 1209 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 19 Except in the field of formal logic almost all traditional proofs in math ematics are really not proofs at all but rather proof outlines or hints as to how to go about constructing the proof Many gaps are left for the reader to fill in There are several reasons for this First it is usually assumed in mathematical literature that the person reading the proof is a mathemati cian familiar with the specialty being described and that the missing steps are obvious to such a reader or at least that the reader is capable of filling them in This attitude is fine for professional mathematicians in the spe cialty but unfortunately it often has the drawback of cutting off the rest of the world including mathematicians in other specialties from understand ing the proof We discussed one possible resolution to this on p x Second it is often assumed that a complete formal proof would require countless millions of symbols Section 1 1 3
289. that the backtracking has not exceeded the specified depth Warn ing Try depth 1 then 2 then 3 etc in sequence because of possible exponential blowups Save your work before trying depth greater than 1 no_distinct Skip trial statements that have d requirements This qualifier will prevent assignments that might violate d requirements but it also could miss possible legal assignments See also set search_limit 5 6 13 save new proof Command Syntax save new_proof label normal compressed The save new_proof command is available in the Proof Assistant only It saves the proof in progress in the source buffer save new_proof may be used to save a completed proof or it may be used to save a proof in progress in order to work on it later If an incomplete proof is saved any user assignments with let step or let variable will be lost as will any ambiguous unifications that were resolved manually To help make recovery easier it can be helpful to improve all before save new_proof so that the incomplete proof will have as much information as possible Note that the proof will not be permanently saved until a write source command is issued Optional command qualifiers normal The proof is saved in the normal format i e as a sequence of labels which is the defined format of the basic Metamath language This 5 7 CREATING BTpxX OUTPUT 147 is the default format that is used if a qualifier is omitted compressed The pro
290. there are computer algebra programs that can do mathemat ics and rather impressively They can instantly solve the integrals that you struggled with in freshman calculus and do much much more But when you look at these programs what you see is a big collection of algorithms and techniques that evolved and were added to over time along with some 8 CHAPTER 1 INTRODUCTION basic software that manipulates symbols Each algorithm that is built in is the result of someone s theorem whose proof is omitted you just have to trust the person who proved it and the person who programmed it in and hope there are no bugs Somehow this doesn t seem to be the essence of mathematics Although computer algebra systems can generate theorems with amazing speed they can t actually prove a single one of them After some puzzlement you revisit some popular books on what mathe matics is all about Somewhere you read that all of mathematics is actually derived from something called set theory This is a little confusing be cause no where in the book that presented the axioms of arithmetic was there any mention of set theory or if there was it seemed to be just a tool that helps you describe things better the set of even numbers that sort of thing If set theory is the basis for all mathematics then why are additional axioms needed for arithmetic Something is wrong but you re not sure what One of your friends is a pure mathematician He kno
291. there are no hidden or built in rules that may be prone to subtle bugs The basic algorithm at the heart of Metamath is simple and fixed and it can be assumed to be bug free and robust with a degree of confidence approaching certainty An independently written implementation of Metamath could pretty much eliminate any residual doubt on the part of a skeptic 1 2 2 Trusting the Mathematician There is no Algebraist nor Mathematician so expert in his sci ence as to place entire confidence in any truth immediately upon his discovery of it or regard it an any thing but a mere probabil ity Every time he runs over his proofs his confidence encreases but still more by the approbation of his friends and is rais d to its utmost perfection by the universal assent and applauses of the learned world DAVID Hume4 Stanislaw Ulam estimates that mathematicians publish 200 000 theorems every year A number of these are subsequently con tradicted or otherwise disallowed others are thrown into doubt and most are ignored RICHARD DE MILLO ET ALP Whether or not the computer can be trusted humans of course will oc casionally err Only the most memorable proofs get independently verified and of these only a handful of truly great ones achieve the status of being known mathematical truths that are used without giving a second thought to their correctness There are many famous examples of incorrect theorems and proofs in mathematical
292. these systems see 31 Since our use for logic is as a basis for set theory which has a non empty domain it is more convenient and more traditional to use a less general system An interesting curiosity is that using a free logic as a basis for Zermelo Fraenkel set theory with the redundant Axiom of the Null Set omitted we cannot even prove the existence of a single set without assuming the axiom of infinity Chapter 2 Using the Metamath Program 2 1 Installation The way that you install Metamath on your computer system will vary for different computers Current instructions are provided with the Metamath program download at In general the installation is simple There is one file containing the Metamath program itself This file is usually called metamath or metamath zzz where vzg is the convention such as exe for an executable program on your operating system There are several additional files containing samples of the Metamath language all ending with mm The file set mm contains logic and set theory and can be used as a starting point for other areas of mathematics You will also need a text editor capable of editing plain ascif text in order to prepare your input files Most computers have this capability built in Note that plain text is not necessarily the default for some word pro cessing programs especially if they can handle different fonts for example with Microsoft Word you must save the file in the format T
293. thout the axioms how do you know you won t stumble across one accidentally when using the informal approach And nowhere do these books describe how all of mathematics can be derived from set theory which by now you ve heard a few times You find a more advanced book on set theory This one actually lists the axioms of ZF set theory in plain English on page one Now you think your quest has ended and you ve finally found the source of all mathematical knowledge you just have to understand what it means Here in one place 13Russell s paradox assumes that there exists a set S that is a collection of all sets that don t contain themselves Now either S contains itself or it doesn t If it contains itself it contradicts its definition But if it doesn t contain itself it also contradicts its definition Russell s paradox is resolved in ZF set theory by denying that such a set S exists 10 CHAPTER 1 INTRODUCTION is the basis for all of mathematics You stare at the axioms in awe puzzle over them memorize them hoping that if you just meditate on them long enough they will become clear Of course you haven t the slightest idea how the rest of mathematics is derived from them in particular if these are the axioms of mathematics then why do arithmetic group theory and so on need their own axioms You start reading this advanced book carefully pondering the meaning of every word because by now you re r
294. til no more unknown steps remain The Proof Assistant will not let you make an assignment unless it can be uni fied with the unknown step This means that a substitution of variables exists that will make the assignment match the unknown step On the other hand in the middle of a proof when working backwards often more than one unification set of substitutions is possible since there is not enough information available at that point to uniquely establish it In this case you can tell Metamath which unification to choose or you can continue to assign unknown steps until enough information is available to make the unification unique We will assume you have entered Metamath and read in the database as described above The following dialog shows how the proof was developed For more details on what some of the commands do refer to Section 5 6 MM gt prove thi Entering the Proof Assistant Type HELP for help EXIT to exit You will be working on the proof of statement thi p t t Note The proof you are starting with is already complete MM PA gt The MM PA gt prompt means we are inside of the Proof Assistant Most of the regular Metamath commands show statement etc are still available if you need them MM PA gt delete all The entire proof was deleted We have deleted the whole proof so we can start from scratch MM PA gt show new_proof lemmon all 17 t t The show new_proof command is like show proof exce
295. tions p statements are derived Their most obvious use is for specifying ordinary mathematical axioms but they are also used for two other purposes First Metamath needs to know the syntax of symbol sequences that con stitute valid mathematical statements A Metamath proof must be broken down into much more detail than ordinary mathematical proofs that you may be used to thinking of even the complete proofs of formal logic This is one of the things that makes Metamath a general purpose language independent of any system of logic or even syntax If you want to use a sub stitution instance of an assertion as a step in a proof you must first prove that the substitution is syntactically correct or if you prefer you must con struct it showing for example that the expression you are substituting for a wff metavariable is a valid wff The a statement is used to specify those combinations of symbols that are considered syntactically valid such as the legal forms of wffs Second a statements are used to specify what are ordinarily thought of as definitions i e new combinations of symbols that abbreviate other combinations of symbols Metamath makes no distinction between axioms and definitions Indeed it has been argued that such distinction should not be made even in ordinary mathematics see Section 4 5 which discusses the philosophy of definitions Section 3 4 discusses some technical requirements 108 CHAPTER 4 THE M
296. tions connect two expressions with either or Eliminating such a definition is a simple matter of substituting the expression on the left hand side definiendum or thing being defined with the equivalent more primitive expression on the right hand side definiens or definition Often a definition has variables on the right hand side which do not ap pear on the left hand side these are called dummy variables In this case any allowable substitution such as a new distinct variable can be used when the definition is eliminated Dummy variables may be used only if they are effectively bound meaning that the definition will remain logically equivalent upon any substitution of a dummy variable with any other quali fying expression i e any symbol string such as another variable that meets the restrictions on the dummy variable imposed by d and f statements For example we could define a constant L inverted tee meaning logical false as p A 7 i e phi and not phi Here is effectively bound because the definition remains logically equivalent when we replace y with any other wff We do not define L in set mm There are two cases where eliminating definitions is a little more complex These cases are the definitions df bi and df cleq The first stretches the concept of a definition a little as in effect it defines a definition however it meets our requirements for a definition in that it is eliminable and does
297. to some arbitrary variable that may occur in the wff The conditional clauses do of course have a precise meaning but as it turns out the precise meaning is somewhat complicated and awkward to formalize in a way that a computer handle easily Unlike propositional calculus a certain amount of mathematical sophistication and practice is needed to be able to easily grasp and manipulate these concepts correctly We take a different approach in the Metamath database set mm We do not use the primitive notions of free variable and proper substitution at all Instead we use a set of axioms that are almost as simple to manipulate as those of propositional calculus Our axiom system avoids complex primitive notions by effectively embedding the complexity into the axioms themselves As a result we will end up with a larger number of axioms but they are ideally suited for a computer language such as Metamath Section 3 3 shows these axioms We will not elaborate on the free variable and proper substitution concepts here We listed the two axioms above so that you will recognize them when you encounter them in the literature You may consult 19 ch 3 4 as well as many other books for a precise explanation of these concepts If you intend to do serious mathematical work it is wise to become familiar with the traditional textbook approach even though the concepts embedded 3 2 THE AXIOMS FOR ALL OF MATHEMATICS 61 in their
298. uch as y aly 176 APPENDIX C METAMATH S FORMAL SYSTEM computer A Metamath database being a finite set of ASCII characters can usually describe only a subset of a formal system and its universe which are typically infinite However the database can contain as large a finite subset of the formal system and its universe as we wish Of course a Metamath set theory database can in principle indirectly describe an entire infinite formal system by formalizing the expository language in this Appendix For purpose of our discussion we assume the Metamath database is in the simple form described on p consisting of all constant and variable declarations at the beginning followed by a sequence of extended frames each delimited by and Any Metamath database can be converted to this form as described on p 114 The math symbol tokens of a Metamath source file which are declared with c and v statements are names we assign to representatives of CN and VR For definiteness we could assume that the first math symbol declared as a variable corresponds to vo the second to v1 etc although the exact correspondence we choose is not important In the Metamath language each d f and e source statement in an extended frame Section 4 2 7 corresponds respectively to a member of the collections D T and H in a formal system statement Dm Tm H A The math symbol strings following these Metamath keywords correspond to a variable pair
299. uded files may include other files which may in turn include other files and so on For example suppose you want to use the set theory database as the starting point for your own theory The first line in your file could be set mm 4 4 EXTENSIONS TO THE METAMATH LANGUAGE 123 All of the information axioms theorems etc in set mm and any files that it includes will become available for you to reference in your file This can help make your work more modular A drawback to including files is that if you change the name of a symbol or the label of a statement you must also remember to update any references in any file that includes it The naming conventions for included files are the same as those of your operating system 9 For compatibility among operating systems you should keep the file names as simple as possible A good convention to use is file mm where file is eight characters or less in lower case There is no limit to the nesting depth of included files One thing that you should be aware of is that if two included files themselves include a common third file only the first reference to this common file will be read in This allows you to include two or more files that build on a common starting file without having to worry about label and symbol conflicts that would occur if the common file were read in more than once In fact if a file includes itself the self reference will be ignored although of course it would n
300. ued and amazed Apparently a mathematician can grasp as a single concept something that would take a computer a thousand or a million steps to verify and have complete confidence in it Each one of these thousand or million steps must be absolutely correct or else the whole proof is meaningless If you added a million numbers by hand would you trust the result How do you really know that all these steps are correct that there isn t some subtle pitfall in one of these million steps like a bug in a computer program After all you ve read that famous mathematicians have occasionally made mistakes and you certainly know you ve made your 1 1 MATHEMATICS AS A COMPUTER LANGUAGE 9 share on your math homework problems in school You recall the analogy with a computer program Sure you can under stand what a large computer program such as a word processor does as a single high level concept or a small set of such concepts but your ability to understand it in no way ensures that the program is correct and doesn t have hidden bugs Even if you wrote the program yourself you can t really know this most large programs that you ve written have had bugs that crop up at some later date no matter how careful you tried to be while writing them OK so now it seems the reason you can t figure out how to make a computer verify proofs is because each step really corresponds to a million small steps Well you say a computer can do a million
301. ule are not a proper part of propositional calculus but are convenient to use when building a hierarchy of provable statements A provable statement with a nonsense hypothesis such as gt and this same expression as its assertion we consider irrelevant no use can be made of it in proving theorems since there is no way to eliminate the nonsense hypothesis Comment Our use of parentheses in the definition of a wff illustrates how axiomatic statements should be carefully stated in a way that ties in unambiguously with the substitutions allowed by the formal system There are many ways we could have defined wffs for example Polish prefix no tation would have allowed us to omit parentheses entirely at the expense of readability but we must define them in a way that is unambiguous For example if we had omitted parentheses from the definition of p gt w the wif gt y could be interpreted as either py gt y or gt y and 168 APPENDIX C METAMATH S FORMAL SYSTEM would have allowed us to prove nonsense Note that there is no concept of operator binding precedence built into our formal system C 3 2 Example 2 Predicate Calculus with Equality Here we extend Example 1 to include predicate calculus with equality illus trating the use of distinct variable restrictions This system is the same as Tarski s system 6g in except that the axioms of propositional calculus are different but equivalent and
302. ument existing mathematics in an absolutely rigorous computer verifiable way not to aid directly in the creation of new mathematics It also is not a magic solution for learning abstract mathematics although it may be helpful to be able to actually see the implied rigor behind what you are learning from textbooks as well as providing hints to work out proofs that you are stumped on 2 Here the notion of verifying correctness of syntax includes verification that a sequen tial list of proof steps results in the specified theorem xii PREFACE As of this writing a sizable set theory database has been developed to provide a foundation for many fields of mathematics but much more work would be required to develop useful databases for specific fields Metamath knows no math it just provides a framework in which to express mathematics Its language is very small You can define two kinds of symbols constants and variables The only thing Metamath knows how to do is to substitute strings of symbols for the variables in an expression based on instructions you provide it in a proof subject to certain constraints you specify for the variables Even the decimal representation of a number is merely a string of certain constants digits which together in a specific context correspond to whatever mathematical object you choose to define for it unlike other computer languages there is no actual number stored inside the computer In a proof
303. urce file May not be used with any other qualifier statement_summary Summarizes all statements like a brief show statement used by the proof It may not be used with any other qual ifier except essential detailed_step step Shows the details of what is happening at a specific proof step May not be used with any other qualifier The step is the step number shown when displaying a proof without the renumber qualifier 5 5 2 show usage Command Syntax show usage label recursive This command lists the statements whose proofs make direct reference to the statement specified Optional command qualifier 138 CHAPTER 5 THE METAMATH PROGRAM recursive Also include statements whose proof ultimately depend on the statement specified 5 5 3 show trace back Command Syntax show trace_back essential axioms tree depth number This command lists all statements that the proof of the specified p statement depends on Optional command qualifiers essential Restrict the trace back to e hypotheses of proof trees axioms List only the axioms that the proof ultimately depends on tree Display the trace back in an indented tree format depth number Restrict the tree trace back to the specified indenta tion depth count_steps Counts the number of steps the proof has all the way back to axioms If essential is specified expansions of variable type hypotheses syntax constructions are not counted
304. used for this purpose and the development of a special axiom system for the non standard formal system becomes unnecessary The model or proof theoretic approach often allows you to prove much deeper results with less effort 1 4 3 Metamath and Its Philosophy Closely related to Metamath is a philosophy or way of looking at mathemat ics This philosophy is related to the formalist philosophy of Hilbert and his followers 27 pp 1203 1208 p 6 In this philosophy mathematics is viewed as nothing more than a set of rules that manipulate symbols to gether with the consequences of those rules While the mathematics being described may be complex the rules used to describe it the metamath ematics should be as simple as possible In particular proofs should be restricted to dealing with concrete objects the symbols we write on paper rather than the abstract concepts they represent in a constructive manner these are called finitary proofs 54 pp 2 3 Whether or not you find Metamath interesting or useful will in part de pend on the appeal you find in its philosophy and this appeal will probably depend on your particular goals with respect to mathematics For exam ple if you are a pure mathematician at the forefront of discovering new mathematical knowledge you will probably find that the rigid formality of Metamath stifles your creativity On the other hand we would argue that once this knowledge is discovered ther
305. used to identify Metamath statements for later reference Label tokens may contain only letters digits and the three characters pe riod hyphen and underscore A label is declared by placing it immediately before the keyword of the statement it identifies For example the label axiom 1 might be declared as follows axiom 1 a x x Each e f a and p statement in a database must have a label declared for it No other statement types may have label declarations Every label must be unique A label and the statement it identifies is referenced by including the label between the and keywords in a p statement The sequence of labels between these two keywords is called a proof An example of a statement with a proof that we will encounter later Section 4 3 is TTo eliminate ambiguity with other expressions this is expressed in the set theory database set mm as 2 2 4 whose IATRX equivalent is 2 2 4 The H means is a theorem and the parentheses allow explicit associative grouping 4 2 THE BASIC KEYWORDS 99 wnew p wif s gt r gt p ws wr wp w2 w2 You don t have to know what this means just yet but you should know that the label wnew is declared by this p statement and that the labels ws wr wp and w2 are assumed to have been declared earlier in the database and are referenced here 4 2 2 Constants and Variables An expression is any sequence of math symbols poss
306. value of the argument Some commands have one or more optional qualifiers which modify the behavior of the command Qualifiers are preceded by a slash such as in read set mm verify Spaces are optional around the If you need to use a slash in a command argument as in a Unix file name put single or double quotes around the command argument The open log command will save everything you see on the screen and is useful to help you recover should something go wrong in a proof or if you want to document a bug If a command responds with more than a screenful you will be prompted to lt return gt to continue Q to quit or S to scroll to end Qorq not case sensitive will complete the command internally but will suppress further output until the next MM gt prompt s will suppress further pausing until the next MM gt prompt After the first screen you are also presented with the choice of b to go back a screenful Note that b may also be entered at the MM gt prompt immediately after a command to scroll back through the output of that command A command line enclosed in quotes is executed by your operating system See Section 5 2 12 Warning Pressing CTRL C will abort the Metamath program uncondi tionally This means any unsaved work will be lost 5 2 1 exit Command Syntax exit force This command exits from Metamath If there have been changes to the source with the save proof or save new_proof commands you will be given a
307. ve In other words submit commands are not allowed inside of a command file 5 2 5 erase Command Syntax erase This command will reset Metamath to its starting state deleting any database that was read in If there have been changes to the source with the save proof or save new_proof commands you will be given an oppor tunity to write source to permanently save the changes 5 2 6 set echo Command Syntax set echo on or set echo off The set echo on command will cause command lines to be echoed with any abbreviations expanded While learning the Metamath commands this feature will show you the exact command that your abbreviated input cor responds to 5 2 7 set scroll Command Syntax set scroll prompted or set scroll continuous The Metamath command line interface starts off in the prompted mode which means that you will prompted to continue or quit after each full screen in a long listing In continuous mode long listings will be scrolled without pausing 5 2 8 set width Command Syntax set width number 5 2 CONTROLLING METAMATH 133 Metamath assumes the width of your screen is 79 characters chosen because the Command Prompt in Windows XP has a wrapping bug at column 80 If your screen is wider or narrower this command allows you to change this default screen width A larger width is advantageous for logging proofs to an output file to be printed on a wide printer A smaller width may be necessary on some terminals i
308. ve terms of sequences to prevent ambiguity and if you examine them with care you will be able to distinguish those parentheses that denote constant symbols from those of our expository language that delimit function arguments Although it might have been better to use boldface for our primitive symbols unfortunately boldface was not available for all characters on the XTEX system used to typeset this text These seemingly forbidding axioms can be understood by analogy to concatenation of substrings in a computer language They are actually relatively simple for each specific case and will become clearer by looking at the special case of a binary predicate a R where rnk R 2 Letting Vs be the sequence x y z the axioms we would add to T for this case would be the wff extension and two equality axioms that are the reducts of the pre statements a T wif Rry T F x z gt Rry gt Rzy a T F y z Ray gt Rzz Study these carefully to see how the general axioms above evaluate to them In practice typically only a few special cases such as this would be needed and in any case the Metamath language will only permit us to describe a finite number of predicates as opposed to the infinite number permitted by the formal system If an infinite number should be needed for some reason we could not define the formal system directly in the Metamath language but could instead define it metalogically und
309. w A rigorous justification for this system and its metalogical completeness is found in 84 The system is closely related to a system developed by Monk and Tarski in 1965 39 For example the formula Jz x y given y there exists some x equal to it is a theorem of logic whether or not x and y are distinct variables In many systems of logic we would have to prove two theorems to arrive at this e result First we would prove Jax x then we would separately prove Axx y where x and y are distinct variables We would then combine these two special cases outside of the system i e in our heads to be able to claim Jaa y regardless of whether x and y are distinct In other words in the combination of the two special cases is a metatheorem In the system of logic used in Metamath s set theory database the axioms of logic are broken down into small pieces that allow them to reassembled in such a way that theorems such as these can be proved directly Breaking down the axioms in this way makes them look peculiar and not very intuitive at first but rest assured that they are correct and complete Their correctness is ensured because they are theorem schemes of standard first order logic which you can easily verify if you are a logician Their completeness follows from the fact that we explicitly derive the standard axioms of first order logic as theorems Deriving the standard axioms is somewhat tric
310. way of knowing whether your axioms are 4 2 THE BASIC KEYWORDS 105 correct For example Metamath would have no idea that ax 16 which we are telling it is an axiom of logic would lead to contradictions if we omitted its associated d statement Comment You may wonder if it is possible to develop standard mathe matics in the Metamath language without the d statement since it seems like a nuisance that complicates proof verification The d statement is not needed in certain subsets of mathematics such as propositional calculus However dummy variables and their associated d statements are impossi ble to avoid in proofs in standard first order logic as well as in the variant used in set mm In fact there is no upper bound to the number of dummy variables that might be needed in a proof of a theorem of first order logic containing 3 or more variables as shown by H Andr ka 43 A first order system that avoids them entirely is given in 34 the trick there is simply to embed harmlessly the necessary dummy variables into a theorem being proved so that they aren t dummy anymore then interpret the result ing longer theorem so as to ignore the embedded dummy variables If this interests you the system in set mm obtained from ax 1 through ax 15 in set mm and deleting ax 16 and ax 17 requires no d statements but is log ically complete in the sense described in 34 This means it can prove any theorem of first order logic as long
311. with an error message if no substitution is possible that will make a referenced assertion s hypothesis match the stack entries You may want to experiment with the different kinds of errors that Metamath will detect by making some small changes in the proof of our example Metamath s proof notation was designed primarily to express proofs in a relatively compact manner not for readability by humans Metamath can display proofs in a number of different ways with the show proof command The lemmon qualifier displays it in a format that is easier to read when the proofs are short and you saw examples of its use in Chapter 2 For longer proofs it is useful to see the tree structure of the proof A tree structure is displayed when the lemmon qualifier is omitted You will probably find this display more convenient as you get used to it The tree display of the proof in our example looks like this 1 wp ws f wff s 2 wp wr f wif r 3 wq wp f wff p 4 wq w2 a wif r gt p 5 wnew w2 a wff s gt r gt p 118 CHAPTER 4 THE METAMATH LANGUAGE The number to the left of each line is the step number Following it is a hypothesis association consisting of two labels separated by To the left of the except in the last step is the label of a hypothesis of an assertion referenced later in the proof here steps 1 and 4 are the hypothesis associations for the assertion w2 that is referenced in step 5 A hypothesis association is
312. with save proof statement compressed During error checking with the verify proof command an error found in the a compressed proof may point to a character in compressed proof which may not be very meaningful to you In this case try to save proof normal first then do the verify proof again In general it is best to make sure a proof is correct before saving it in compressed format because severe errors are less likely to be recoverable than in normal format 4 4 5 Specifying Unknown Proofs or Subproofs In a proof under development any step or subproof that is not yet known may be represented with a single For the purposes of parsing the proof the will push a single entry onto the RPN stack just as if it were a hypothesis While developing a proof with the Proof Assistant a partially developed proof may be saved with the save new_proof command and s will be placed at the appropriate places All p statements must have proofs even if they are entirely unknown Before creating a proof with the Proof Assistant you should specify a com pletely unknown proof as follows label p statement The verify proof command will check the known portions of a partial proof for errors but will warn you that the statement has not been proved Note that partially developed proofs may be saved in compressed format if desired In this case you will see one or more s in the compressed proof part 4 5 APPENDIX AXIOMS VS DE
313. ws he is unable to communicate to you what he does for a living and seems to have little interest in trying You do know that for him proofs are what mathematics is all about You ask him what a proof is and he essentially tells you that while of course it s based on logic really it s something you learn by doing it over and over until you pick it up He refers you to a book How to Read and Do Proofs 55 Although this book helps you understand traditional informal proofs there is still something missing you can t seem to pin down yet You ask your friend how you would go about having a computer verify a proof At first he seems puzzled by the question why would you want to do that Then he says it s not something that would make any sense to do but he s heard that you d have to break the proof down into thousands or even millions of individual steps to do such a thing because the reasoning involved is at such a high level of abstraction He says that maybe it s something you could do up to a point but the computer would be completely impractical once you get into any meaningful mathematics There the only way you can verify a proof is by hand and you can only acquire the ability to do this by specializing in the field for a couple of years in grad school Anyway he thinks it all has to do with set theory although he has never taken a formal course in set theory but just learned what he needed as he went along You are intrig
314. x is a natural number i e the set of natural numbers here x is a natural number is a rather complicated formula when 3 1 LOGIC AND SET THEORY 57 broken down into the primitive symbols Actually the primitive symbols don t even include the brace notation The brace notation is a high level definition which you can find in Section 3 4 Interestingly the arithmetic of integers and rationals can be developed without appealing to the existence of an infinite set whereas the arithmetic of real numbers requires it Each variable in the axioms of set theory represents an arbitrary set and the axioms specify the legal kinds of things you can do with these variables at a very primitive level Now you may think that numbers and arithmetic are a lot more intu itive and fundamental than sets and therefore should be the foundation of mathematics What is really the case is that you ve dealt with numbers all your life and are comfortable with a few rules for manipulating them such as addition and multiplication Those rules only cover a small portion of what can be done with numbers and only a very tiny fraction of the rest of mathematics If you look at any elementary book on number theory you will quickly become lost if these are the only rules that you know Even though such books may present a list of axioms for arithmetic the ability to use the axioms and to understand proofs of theorems facts about numbers requires an imp
315. xcept that there is no statement argument since it is the statement being proved and following qualifiers are not available statement_summary detailed_step Also the following additional qualifiers are available unknown Shows only steps that have no statement assigned not_unified Shows only steps that have not been unified Note that essential depth unknown and not_unified may be used in any combination each of them effectively filters out additional steps from the proof display See also show proof 5 6 CREATING PROOFS 143 5 6 6 assign Command Syntax assign step label and assign first label and assign last label This command available in the Proof Assistant only assigns an unknown step one with inthe show new_proof listing with the statement specified by label The assignment will not be allowed if the statement cannot be unified with the step If last is specified instead of step number the last step that is shown by show new_proof unknown will be used This can be useful for building a proof with a command file see help submit It also makes building proofs faster when you know the assignment for the last step If first is specified instead of step number the first step that is shown by show new_proof unknown will be used If step is zero or negative the stepth from last unknown step as shown by show new_proof unknown will be used assign 1 label will assign the penultimate unknown s
316. xpression symbol sequence for a variable subject to a simple constraint to prevent bound variable clashes The primitive notions built into Metamath involve the simple manipulation of finite objects symbols that we as humans can eas ily visualize and that computers can easily deal with They seem to be just about the simplest notions possible that are required to do standard mathematics This chapter serves as a reference manual for the Metamath language It covers the tedious technical details of the language some of which you may wish to skim in a first reading On the other hand you should pay close attention to the defined terms in boldface they have precise meanings that are important to keep in mind for later understanding It may be best to first become familiar with the examples in Chapter 2 to gain some motivation for the language If you have some knowledge of set theory you may wish to study this chapter in conjunction with the formal set theoretical description of the Metamath language in Appendix C We will use the name Metamath to mean either the Metamath com puter language or the Metamath software associated with the computer 1 52 p 84 91 92 CHAPTER 4 THE METAMATH LANGUAGE language We will not distinguish these two when the context is clear The next section contains the complete specification of the Metamath language It serves as an authoritative reference and presents the syntax in enough d
317. xtbooks are usually written to train mathematicians or to communicate to people who are already mathematicians and large gaps in proofs are often left as exercises to the reader who is left at an impasse if he or she becomes stuck I believe that eventually computers will enable non specialists and even intelligent laypersons to follow almost any mathematical proof in any field Metamath is an attempt in that direction If all of mathematics were as eas ily accessible as a computer programming language I could envision com puter programmers and hobbyists who otherwise lack mathematical sophis tication exploring and being amazed by the world of theorems and proofs in obscure specialties perhaps even coming up with results of their own A tremendous advantage would be that anyone could experiment with conjec tures in any field the computer would offer instant feedback as to whether an inference step was correct Mathematicians sometimes have to put up with the annoyance of cranks who lack a fundamental understanding of mathematics but insist that their proofs of say Fermat s Last Theorem be taken seriously I think part of the problem is that these people are mislead by informal mathematical language treating it as if they were reading ordinary expository English and failing to appreciate the implicit underlying rigor Such cranks are rare in the field of computers because computer languages are much more explicit and ultimately the
318. y You re in for a surprise you can conceive of no way to convert the proofs which are in English to a form that the computer can understand The proofs are filled with phrases such as assume there exists a unique xz and given any y let z be the number such that This isn t the kind of logic you are used to in computer programming where everything even arithmetic reduces to Boolean ones and zeroes if you care to break it down sufficiently Even though you think you understand the proofs there seems to be some kind of higher reasoning involved rather than precise rules that define how you manipulate the symbols in the axioms Whatever it is it just isn t obvious how you would express it to a computer and the more you think about it the more puzzled and confused you get to the point where you even wonder whether you really understand it There s a lot more to these axioms of arithmetic than meets the eye Nobody ever talked about this in school in your applied math and en gineering courses You just learned the rules they gave you not quite un derstanding how or why they worked sometimes vaguely suspicious or un certain of them and through homework problems and osmosis learned how to present solutions that satisfied the instructor and earned you an A Rarely did you actually prove anything in a rigorous way and the math majors who did do stuff like that seemed to be in a different world Of course
319. y may not be simultaneously substituted with the same variable The statement d xy means variable x must not occur in wff y For the precise definition of d see Section 4 2 4 In our system the axioms of predicate calculus have been divided into what we call pure predicate calculus which uses no or symbols and equality and substitution The former is weaker than traditional predicate calculus because it does not have substitution although it is a beautifully simple and interesting system in itself and the latter extends its power to the full predicate calculus with equality You should also note that our system of predicate calculus is specifically tailored for set theory thus there are only two specific predicates and 64 CHAPTER 3 ABSTRACT MATHEMATICS REVEALED and no functions or constants unlike more general systems Finally I do not claim that these axioms are the most elegant possible They have more metalogical power than standard axioms although they are equivalent in a logical sense and the ones I have chosen at least do the job Perhaps a logician who reads this would be interested in devising a set of equivalent axioms that are shorter or more elegant in some sense For example there used to be an axiom ax 15 in our list that was later dis covered to be redundant in this version of the axiom system The Comment on p 105 has a further remark on this axiom A rigorous justification f
320. ystem at all but rather as abbreviations that are part of the expository metalogic outside the formal system For convenience though we may use the formal C 3 EXAMPLES OF FORMAL SYSTEMS 173 system itself to incorporate definitions adding them as axiomatic exten sions to the system This could be done by adding a constant representing the concept is defined as along with axioms for it But there is a nicer way at least in this writer s opinion that introduces definitions as direct extensions to the language rather than as extralogical primitive notions We introduce additional logical connectives and provide axioms for them For systems of logic such as Examples 1 through 3 the additional axioms must be conservative in the sense that no wff of the original system that was not a theorem when the initial term wff is replaced by of course becomes a theorem of the extended system In this example we extend Example 3 or 2 with standard abbreviations of logic We extend CN of Example 3 with new constants 4 A V corre sponding to logical equivalence conjunction disjunction and the existential quantifier We extend I with the axiomatic statements that are the reducts of the following pre statements T wit y e w S T wif yp V T wE p A w o T wif Ax y T H ine he ae 2 T 2 F p 4 gt U gt 9 T H a p W gt p

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