http://dx.doi.org/10.1090/coll/041

AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS

VOLUME 41

A FORMALIZATION OF SET THEORY WITHOUT VARIABLES

BY ALFRED TARSKI

and STEVEN GIVANT

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1985 Mathematics Subject Classification. Primar y 03B; Secondary 03B30 , 03C05, 03E30, 03G15.

Library o f Congres s Cataloging-in-Publicatio n Dat a

Tarski, Alfred . A formalization o f se t theor y withou t variables .

(Colloquium publications , ISS N 0065-9258; v. 41) Bibliography: p. Includes indexes. 1. Se t theory. 2 . Logic , Symboli c an d mathematical . I . Givant, Steve n R . II. Title. III. Series: Colloquium publications (American Mathematical Society) ; v. 41. QA248.T37 198 7 511.3'2 2 86-2216 8 ISBN 0-8218-1041-3 (alk . paper )

Copyright © 198 7 b y th e America n Mathematica l Societ y Reprinted wit h correction s 198 8 All rights reserve d excep t thos e grante d t o th e Unite d State s Governmen t This boo k ma y no t b e reproduce d i n an y for m withou t th e permissio n o f th e publishe r The pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . @ Contents

Section interdependenc e diagram s vii Preface x i

Chapter 1 . Th e Formalis m £ o f Predicate Logi c 1 1.1. Preliminarie s 1 1.2. Symbol s an d expression s o f £ 4 1.3. Derivabilit y i n £ 7 1.4. Semantica l notion s o f £ 1 1 1.5. First-orde r formalism s 1 4 1.6. Formalism s an d system s 1 6

Chapter 2 . Th e Formalis m £+, a Definitional Extensio n o f £ 2 3 2.1. Symbol s an d expression s o f L + 2 3 2.2. Derivabilit y an d semantica l notion s o f L + 2 5 2.3. Th e equipollenc e o f L + an d L 2 7 2.4. Th e equipollenc e o f a system wit h a n extensio n 3 0 2.5. Th e equipollenc e o f two systems relativ e t o a commo n extension 4 1

Chapter 3 . Th e Formalis m L x withou t Variable s an d th e Problem o f Its Equipollenc e wit h £ 4 5 3.1. Syntactica l an d semantica l notion s o f £x 4 5 3.2. Schemat a o f equations derivabl e i n £x 4 8 3.3. A deduction theore m fo r £x 5 1 3.4. Th e inequipollenc e o f £x wit h £+ an d £ 5 3 3.5. Th e inequipollenc e o f extensions o f £x wit h £+ an d £ 5 6 3.6. £ x-expressibility 6 2

3.7. Th e three-variabl e formalism s £ 3 an d £+3 " 6 4 3.8. Th e equipollenc e o f £3 an d £3 " 7 2 iV CONTENTS

3.9. Th e equipollenc e o f £x an d £^ 7 6 3.10. Subformalism s o f £ an d £+ wit h finitely man y variable s 8 9

Chapter 4 . Th e Relativ e Equipollenc e o f £ an d £x, an d th e Formalization o f Set Theor y i n £ x 9 5 4.1. Conjugate d quasiprojection s an d sentence s QAB 9 5 4.2. System s o f conjugated quasiprojection s an d system s o f predicates PAB 10 0 4.3. Historica l remark s regardin g the translation mappin g fro m £+ t o £x 10 7 4.4. Proo f o f the main mappin g theore m fo r £x an d £+ 11 0 4.5. Th e constructio n o f equipollent Q-system s i n £x 12 4 4.6. Th e formalizabilit y o f systems o f set theor y i n £x 12 7 4.7. Problem s o f expressibility an d decidabilit y i n £x 13 5 4.8. Th e undecidabilit y o f first-order logic s with finitely man y variables, and the relativ e equipollenc e o f £3 wit h £ 14 0

Chapter 5 . Som e Improvements o f the Equipollence Result s 14 7 5.1. One-on e translatio n mapping s 14 7 5.2. Reducin g th e number o f primitive notion s o f £ x: definitionally equivalen t variant s o f £x 15 1 5.3. Eliminatin g th e symbo l 1 as a primitive notio n fro m systems o f set theor y i n £x 15 3 5.4. Eliminatin g th e symbo l = a s a primitive notio n fro m £ x: the reduce d formalis m £x 15 8 5.5. Undecidabl e subsystem s o f sentential logi c 16 5

Chapter 6 . Implication s o f the Mai n Result s fo r Semanti c an d Axiomatic Foundation s o f Set Theor y 16 9 6.1. Denotatio n an d truth i n £x 16 9 6.2. Th e denotabilit y o f first-order definabl e relation s i n O-structures 17 0 6.3. Th e £ x-expressibility o f certain relativize d sentence s 17 4 6.4. Th e finite axiomatizabilit y o f predicative system s o f set theory admittin g prope r classe s 17 7 6.5. Th e finite axiomatizabilit y o f predicative system s o f set theory excludin g proper classe s 18 7

Chapter 7 . Extensio n o f Results to Arbitrary Formalism s o f Predicat e Logic, and Application s t o the Formalizatio n o f the Arithmetic s o f Natural an d Rea l Numbers 19 1 7.1. Extensio n o f equipollence result s to Q-system s i n first-order formalism s wit h just binar y relatio n symbol s 19 1 CONTENTS v

7.2. Extensio n o f equipollence result s t o wea k Q-system s i n arbitrary first-orde r formalism s 20 0 7.3. Th e equipollenc e o f weak Q-system s wit h finit e variabl e subsystems 20 8 7.4. Compariso n o f equipollence result s fo r stron g an d wea k Q-systems 21 4 7.5. Th e formalizabilit y o f the arithmeti c o f natural number s in £x 21 5 7.6. Th e formalizabilit y o f Peano arithmeti c i n £x, an d th e definitional equivalenc e o f Peano arithmeti c wit h a syste m of set theor y 22 2 7.7. Th e formalizabilit y o f the arithmeti c o f real numbers i n £x 22 6 7.8. Remark s o n first-order formalism s wit h limite d vocabularies 22 9 Chapter 8 . Application s t o Relation Algebra s an d t o Varietie s o f Algebras 23 1 8.1. Equationa l formalism s 23 1 8.2. Relatio n algebra s 23 5 8.3. Represen t able relation algebra s 23 9 8.4. Q-relatio n algebra s 24 2 8.5. Decisio n problem s fo r varietie s o f relation algebra s 25 1 8.6. Decisio n problem s fo r varietie s o f groupoids 25 8 8.7. Historica l remark s regardin g the decisio n problem s 26 8 Bibliography 27 3 Indices Index o f Symbol s 28 3 Index o f Names 29 7 Index o f Subject s 30 1 Index o f Numbered Item s 31 7 This page intentionally left blank Explanation o f section interdependenc e diagram s

The diagram s o n the next tw o pages indicat e th e essentia l interdependencie s of th e variou s section s o f thi s book . I n general , th e dependenc e o f a sectio n on earlie r section s i s determine d b y followin g upward s th e line s leadin g t o th e section's box. Fo r example, Section 4.8 depends on Sections 3.8-3.9 (an d possibly on section s abov e them , suc h a s 3.7 , 3.1-3.3 , 2.1-2.3 , an d 1.2-1.4) , a s wel l a s on Section s 4.1-4. 4 (an d possibl y o n section s abov e them) . A smal l par t o f i t also depends o n part o f Section 3.10 ; this mor e limite d dependenc e i s indicate d by a dotte d line . Fo r a secon d example , Sectio n 4. 7 depend s o n 3. 6 (whic h i n turn depend s o n earlie r sections) , a s wel l a s o n 4. 6 (an d possibl y o n som e o f the section s abov e it , suc h a s 4.1-4.5). A s a final example , Sectio n 6. 2 depend s on 6. 1 an d 4.4 . ( A lin e flows to 6. 2 fro m th e bo x labele d 4.1-4.5 , but w e hav e indicated parentheticall y tha t onl y 4. 4 i s really important. ) A small part o f 6. 2 also uses some results fro m 3.8-3.9 .

Vll Diagrams of Section Interdependenc e

EquipoUence Result s

1.2-1.4 Description o f £

1.5 First-order formalism s

1.6 2.1-2.3 Formalisms an d system s £ + an d it s equipoUenc e with £

2.4-2.5 3.1-3.3 General remark s Description o f £ x on equipoUenc e

3.4 3.5 5.2-5.4 3.6 I x Inequipollence o f Inequipollence o f Alternative Description o f £3 an d £3 " £ -expressibilit y £x wit h £ + an d £ extensions o f £ x formalisms I with £ i n mean s to£x n_ of expressio n 3.8 3.9 EquipoUence o f EquipoUence x £3 an d £3 " of £ an d £ t

4.1-4.4 3.10 Relative equipoUenc e Finite variabl e x of £ an d £ + wit h £ subformalisms of £

4.5 4.8 EquipoUence o f Q-system s Relative equipoUenc e x (in £ ) wit h system s i n £ of £ an d £ 3

5.1 Improvement o f Main Mappin g Theorem

1.5 I 7.1-7.2 EquipoUence o f wea k Q-systems with system s i n £ x

7.3-7.4 EquipoUence o f wea k Q-systems wit h finit e variable subsystem s Applications

8.1 Equational logi c

3.1-3.3

8.2 Relation algebra s 3.1

6.1 8.3 Definition o f Representable truth i n L x relation algebra s 7.1-7.2

3.8-3.9 (4.4) (4.4) (7.1) un

6.2 4.6 8.4 7.5 Characterization o f Formalizability o f Representation an d Formalizability o f th e x first-order definabl e set theorie s i n £ axiomatization problem s arithmetic o f natura l x relations i n H-structure s for classe s o f numbers i n £ Q-relation algebra s

6.3 4.7 7.6 £x-expressibility Expressibility an d Formalizability o f of certain relativize d decidability Peano arithmeti c x sentences in £ x in £

6.4-6.5 5.5 8.5 7.7 Finite axiomatizabilit y Undecidable Decision problem s Formalizability o f of predicative system s subsystems o f for varietie s o f the arithmeti c o f x of set theor y sentential logi c relation algebra s real number s i n £ 03

8.6 Decision problem s fo r varieties o f groupoid s

8.7 Historical remark s regarding th e decision problem s This page intentionally left blank Preface

In thi s wor k w e shal l sho w tha t se t theor y an d numbe r theor y ca n b e de - veloped withi n th e framewor k o f a new , different , an d ver y simpl e formalism , £ x. £ x i s closely relate d t o th e equational theor y o f abstract relatio n algebra s essentially give n in Chin-Tarski [1951] . Its language contains no variables, quan- tifiers, o r sentential connectives. Ther e are two basic symbols, 1 and E, intende d to denote th e identit y an d the (set-theoretic ) membershi p relations. Compoun d expressions ar e constructe d fro m th e basi c symbol s b y mean s o f fou r operatio n symbols, © , w , + , an d ~ , tha t denot e th e well-know n operation s (o n an d t o binary relations ) o f relative product, conversion , Boolea n addition , an d comple - mentation. Al l mathematical statements i n £x ar e formulated a s (variable-free ) equations betwee n suc h expressions . Th e deductiv e apparatu s o f £x i s base d upon te n logica l axiom schemata that ar e the analogues o f the equational postu - lates for abstract relation algebras essentially given in Chin-Tarski [1951] , p. 344. There i s just on e rul e o f inference , namely , th e rul e familia r fro m hig h schoo l algebra o f replacin g equal s b y equals . A (deductive ) syste m i n £x i s give n b y a se t o f nonlogical axioms , i.e. , equation s o f £x, an d ca n be identifie d wit h th e theory i n £x generate d b y these axioms, i.e., with the se t o f all equations deriv - able fro m thes e axioms , th e logica l axiom s o f £x, an d equation s o f th e for m A = A , b y means o f the singl e rule o f replacing equal s b y equals . £ x appear s t o b e quite wea k i n its powers o f expression an d proof . Eve n th e simple statemen t tha t ther e exis t a t leas t fou r element s canno t b e equivalentl y expressed i n £ x, a s follows at onc e from a result o f Korselt give n in Lowenhei m [1915], p. 44 8 (se e below) . Furthermore , £ x i s semantically incomplete i n th e sense that ther e are semantically vali d equations i n £x whic h are not derivable . This follows readily from a result o f McKenzie [1970 ] (sharpening an earlier result of Lyndon [1950] ) by which the postulates fo r abstract relatio n algebra s give n in Chin-Tarski [1951 ] do not eve n yiel d al l o f the equation s wit h just on e variabl e that ar e true i n ever y concret e algebr a o f relations. W e shall show , i n fact, tha t £x i s equipollen t (i n a natural sense ) t o a certai n fragment , £3 , o f first-orde r logic havin g on e binar y predicat e an d containin g just three variables. (£ 3 i s

XI xn PREFACE

also semanticall y incomplete. ) I t i s therefor e quit e surprisin g tha t £ x prove s adequate fo r th e formalizatio n o f practicall y al l know n system s o f se t theory , and henc e fo r the developmen t o f all o f classical mathematics . As a languag e suitabl e fo r th e formalizatio n o f most set-theoretica l systems , we take the first-order logi c L wit h equality and on e nonlogical binary predicat e E. (For technical reason s w e use "1 " instea d o f " = " a s the name o f the symbo l denoting th e relatio n o f equalit y betwee n individuals. ) A syste m i n £ i s give n by a set o f nonlogical axioms , and , a s before , ca n b e identifie d wit h th e theor y in C generated b y these axioms . It proves convenient t o consider als o an auxiliary formalis m £+ tha t i s a kind of definitional extensio n o f £. I n addition t o the basi c predicates 1 and E o f £ , the vocabulary o f £+ contain s as logical constants (o f a new kind) the symbols © , w, + , an d " fro m £x, b y means o f which (compound ) predicates are constructe d from 1 and E\ specifically , i f A an d B ar e predicates , the n s o ar e A©B, v4 w, A + B, and A~. Th e vocabulary o f £+ als o contains the second equality symbol , = , from £x, intende d to denote the relation o f equality between binary relations. Atomic formulas o f £+ ar e expressions of the form xAy an d A = B, where £, y are individual variables and A, B ar e predicates. Arbitrar y formula s ar e constructe d from atomi c one s in the usual way . I n addition t o a set o f logical axiom s simila r in characte r t o thos e o f £ , £ + ha s five axio m schemat a tha t ca n b e regarde d as possibl e definition s o f the constant s © , w, + , ~ , an d = . Fo r example , th e schemata fo r © an d = ar e respectivel y

Vxy(xAoBy <- • 3z(xAz A zBy)), and

A = B ++ Vxy(xAy <- * xBy), where A,B ar e arbitrar y predicate s o f £+. Th e rule s o f inferenc e fo r £+ an d the notio n o f a system i n £+ ar e taken just a s i n £. A "definitiona l extension " of £ whic h essentially include s £+ i s discussed i n Quine [1969] , pp. 15-27, under the nam e o f "th e virtual theor y o f classes". With th e hel p o f £+ w e shall compar e th e power s o f expression an d proo f o f £ an d £ x, an d als o o f system s develope d i n thes e formalisms . Sinc e £x ha s no variables , w e mus t replac e familia r notion s lik e "definitionall y equivalent " by suitabl e analogues . I n eac h o f th e formalism s £ , £ +, an d £ x (and , mor e generally, i n ever y syste m develope d i n thes e formalisms ) ther e i s the notio n o f sentence an d the notio n o f derivability, i.e. , o f a sentence being derivable (i n th e formalism o r system) from a set of sentences. Suppos e Si an d § 2 are formalism s (or systems ) tha t hav e thes e tw o notions . W e sa y tha t § 2 i s a n extension o f §1 i f ever y sentenc e o f § 1 i s a sentenc e o f § 2 an d derivabilit y i n S i implie s derivability i n S2. Suc h an extension S 2 is called an equipollent extension o f Si i f also the followin g tw o conditions hold: (1 ) (equipollenc e i n means o f expression ) for ever y sentenc e X o f S 2 ther e i s a sentenc e Y o f S i tha t i s equivalen t t o

X i n S 2, i.e. , Y i s derivabl e fro m X , an d X fro m y , i n S 2; (2 ) (equipollenc e in mean s o f proof ) fo r ever y sentenc e X an d se t o f sentence s ^ o f Si , i f X PREFACE xm is derivabl e fro m # i n §2 , the n i t i s s o derivabl e i n §1 . (W e avoi d th e ter m "definitional extension " becaus e i t usuall y involve s a versio n o f (1 ) applyin g t o arbitrary formulas , an d no t jus t t o sentences. ) Finally , § 1 an d § 2 ar e sai d t o be equipollent i f they hav e a commo n equipollen t extension . Whe n w e wish t o emphasize th e rol e o f a particula r commo n equipollen t extensio n §3 , w e shal l say tha t § 1 an d § 2 ar e equipollent relative to S3 . I t i s no t difficul t t o sho w that whe n §1 and § 2 ar e equipollent , ther e i s a natural one-on e correspondenc e between th e theorie s (i.e. , th e deductivel y close d set s o f sentences ) i n § 1 an d §2 tha t preserve s variou s importan t propertie s o f theorie s suc h a s consistenc y and completenes s (cf . Theore m 2.4(viii ) below) ; a theory i s consistent i f it doe s not coincid e wit h th e se t o f al l sentences , an d i t i s complete i f i t i s a maxima l consistent se t o f sentences. It i s readil y see n (an d follow s fro m wha t i s i n Quin e [1969] ) tha t £ + i s a n equipollent extensio n o f £ (an d eve n more ; cf . §2.3) . I t i s als o eas y t o sho w (using, e.g. , th e semanti c completenes s o f £+) tha t £ + i s an extensio n o f £ x. However, i t i s no t a n equipollen t extension , an d i n fac t bot h (1 ) an d (2 ) fail . The failur e o f (1 ) i s a direc t consequenc e o f Korselt' s result , cite d abov e (se e Theorem 3.4(iv)) , whil e the failur e o f (2 ) i s due to the aforementione d semanti c incompleteness o f £x (se e Theore m 3.4(vi)) . Regardin g (1) , w e shal l actuall y prove (i n Theorem 3.5(viii) ) th e followin g muc h stronge r result .

(i) Even if we enrich £x (and £ +)by adjoining any finite number of new con- stants, all of which are intended to denote operations on and to binary relations, or else relations between binary relations (over the universe of any realization of £x), and which are "logical" in the sense that the denoted operations and relations are preserved under all permutations of the universe, there will still be sentences of £ that are not equivalent (in the enriched L~*~)with any sentence of the enriched £ x .

Thus, the inadequacy o f the expressive powers of £x i s not due to a faulty choic e of the set o f fundamental notions ; there i s no way o f extending this set i n a finite and "logical " wa y s o a s to achiev e equipollenc e wit h £ i n means o f expression . We shall als o prove (i n §§3. 8 and 3.9 ) th e theorem , referre d t o before , that : (ii) £ x is equipollent to a certain three-variable fragment, £3 , of £ (relative to a similar fragment, £3" , o/£ +). As we have seen, £ x i s weaker than £ bot h i n means o f expression and proof . Nevertheless, as stated above , we are going to establish the surprising result tha t £x i s adequate fo r th e developmen t o f classica l mathematics . Thi s wil l follo w from tw o theorems, (iii ) an d (iv) , which w e now describe . In (iii ) we shall show that fo r certain special systems in £ calle d Q-system s we can construc t equipollen t system s i n £x. A system S in £ i s called a Q-syste m if there are formulas D an d E (o f £) containin g at mos t thre e distinct variables , and just two free variables, such that i n every model of §, the two binary relations defined b y D and E for m a pair o f conjugated quasiprojections , i.e. , are function s XIV PREFACE with th e followin g additiona l property : fo r ever y pai r o f element s x, y (i n th e universe o f the model ) ther e i s a z whic h i s mapped t o x b y th e first functio n and t o y b y the second ; the elemen t z shoul d b e thought o f as representing th e ordered pair (x, y). Ou r main equipollence theorem (whic h is established in §§4.4 and 4.5 , an d upoi * which mos t o f the late r result s i n the boo k ar e based ) i s a s follows.

(Hi) Every Q-system § in £ is equipollent with a system in £x {relative to a system in £+); moreover, the system in £x will be, e.g., finitely axiomatizable or decidable iff S is.

In (iv ) thi s theore m i s generalize d t o weak Q-systems develope d i n arbitrary first-order formalism s wit h finitely man y nonlogica l constants. Th e definitio n o f a weak Q-syste m i s obtained fro m tha t o f a Q-syste m b y dropping the restrictio n on the numbe r o f distinct variable s occurrin g i n formula s D an d E. W e prove, in fac t (i n Theorem 7.2(iv)) , that :

(iv) Every weak Q-system U developed in a first-order formalism with finitely many nonlogical constants is equipollent with a system in L x; again, this latter system is, e.g., finitely axiomatizable or decidable iffU is.

Both (iii ) an d (iv ) see m ver y specialized . However , w e shal l sho w (i n §4.6 ) that the hypothesis o f (iii) holds for almost every known system o f set theory, an d (in §§7.5-7.7 ) tha t th e hypothesi s o f (iv ) applies , e.g. , t o th e (full ) elementar y theory o f natural numbers, to its well-known, recursively axiomatize d subtheory , first-order Pean o arithmetic , an d t o th e elementar y theor y o f the rea l number s (with th e se t o f natural number s a s a distinguished subset) . Thu s eac h o f thes e systems i s equipollent t o a system i n £ x. With the help of the equipollence theorems (ii)-(iv ) w e shall also investigate a variety o f other problems , quite apart fro m th e on e o f formalizing mathematica l systems i n £ x. Thes e concern , fo r example , th e constructio n o f undecidabl e subsystems o f sentential logi c (i n §5.5) , the relativel y simpl e definitio n o f trut h for th e formalis m £ x (i n 6.1(i),(ii)) , a characterizatio n o f th e first-order de - finable binar y relation s i n model s o f se t theor y an d arithmeti c (i n 6.2(ix ) an d 7.4), th e finite axiomatizabilit y o f predicativ e version s o f system s o f se t the - ory (i n 6.4(vi ) an d 6.5(iv)) , th e adequac y o f first-order formalism s wit h onl y finitely man y variable s fo r th e developmen t o f various mathematical discipline s (in 4.8(xi),(xii) , 7.3(h),(iii) , an d 7.5(vi)) , th e first-order definitiona l equivalenc e of number theory and the theory o f hereditarily finite set s (i n 7.5(v) an d 7.6(h)) , the representatio n proble m fo r relatio n algebra s wit h a pai r o f quasiprojectiv e elements, an d th e nonfinit e axiomatizabilit y o f th e equationa l theor y o f thes e algebras (i n 8.4(iii),(vii)) , th e undecidabilit y o f the equationa l theor y o f severa l important classe s of relation algebras (i n 8.5(xii)), and what seems to be the first construction o f a finitely based , essentiall y undecidabl e equationa l theor y (an d in fact a theory o f groupoids—see 8.5(xi ) an d 8.6(x)) . PREFACE xv

We now make some historical remarks regarding the above theorems and thei r relation t o result s i n th e literature . Th e mathematic s o f th e presen t wor k i s rooted i n the calculus o f relations (o r the calculu s o f relatives, as it i s sometimes called) that originated in the work of A. De Morgan, C. S. Peirce, and E. Schroder during th e secon d hal f o f th e nineteent h century . Th e univers e o f discours e o f this calculus i s the collection o f all binary relation s o n an arbitrary bu t fixed se t £/, i.e. , th e se t o f al l subset s o f U x U. Ther e ar e si x fundamenta l operation s on and to (binary ) relations , and fou r distinguishe d relations . Specifically , ther e are th e fou r binar y operation s o f forming , fo r an y tw o relations R an d 5, thei r absolute sum , which i s simply the union R U S, thei r absolut e product, whic h i s the intersection Rf)S, thei r relative sum, it! f S, consistin g o f all pairs (x , y) suc h that fo r every z either xRz o r zSy (i.e. , either (x , z) i s in R o r (z , y) i s in S), an d their relativ e product , R\S, consistin g o f al l pair s (x,y) suc h tha t fo r som e z , both xRz an d zSy. Further , there are two unary operations o f forming, fo r ever y relation R, it s complement , ~/? , with respec t t o U x [/ , an d it s converse , R~ x, consisting o f al l pairs (x,y) suc h that yRx. Finally , th e distinguishe d relation s are the absolute zero , which i s the empty relatio n 0, th e absolute unit, whic h i s the universal relation U xU, th e relative zero , which i s the diversit y relatio n Di on U consisting o f all pairs (x , y) suc h that x ^ y, an d the relative unit, whic h i s the identit y relatio n Id o n U consisting o f al l pairs (x , y) suc h that x — y. Thi s is the framework o f the calculus as finally presented i n Peirce [1882 ] after severa l earlier versions . Both Peirc e and , later, Schroder , wh o extended Peirce' s wor k i n a very thor - ough an d systemati c wa y i n Schrode r [1895] , wer e intereste d i n th e expressiv e powers o f th e calculu s o f relation s an d th e grea t diversit y o f law s tha t coul d be proved . The y wer e awar e that man y elementar y statement s abou t (binary ) relations ca n b e expresse d a s equation s i n thi s calculus . (B y a n "elementar y statement" abou t relation s R,S,... (ove r U) w e mea n a first-order statemen t about th e structure (U, i? , S,...).) T o give an example , th e (elementary ) state - ment tha t a relation R i s transitive, for every x,y,z, if xRy and yRz, then xRz, can b e expressed b y the equatio n (R\R)\JR = R. Similarly, the more complicated statement that R i s a one-one function mappin g U ont o itsel f i s rendered b y the equatio n

[{RIR-1) n Di] U [{R-^R) n Di] U [-# 10 ] U [01 ~R] = 0- Schroder seem s t o hav e bee n th e first t o conside r th e questio n whethe r al l elementary statements about relations are expressible as equations in the calculus of relations, an d i n Schrode r [1895] , p. 551 , he propose d a positiv e solutio n t o the problem. A critique o f Schroder's proposed solutio n appeared i n Lowenhei m [1915], p. 450, along with a negative solution due to Korselt that wa s referred t o XVI PREFACE above. A far-reachin g extensio n o f Korselt's result , formulate d i n (i ) abov e fo r £x (bu t als o true i n the mor e genera l settin g o f the calculu s o f relations), wa s first announce d i n Tarski [1941] . In th e sam e pape r Tarsk i pose d th e proble m o f provin g tha t ther e i s n o al - gorithm fo r decidin g i n ever y particula r cas e whether a n elementar y statemen t about relations is expressible in the calculus (as an equation). Michae l Kwatinet z finally settle d th e problem aroun d 1971 , with the help o f (ii ) abov e (restate d fo r the calculu s o f relations) , b y showin g tha t th e se t o f elementar y statement s which ca n equivalentl y b e formulate d usin g just thre e variable s i s not recursiv e (see Kwatinetz [1981 ] for th e proof) . Despite th e wea k expressiv e power s o f th e calculu s o f relations , Tarsk i wa s able to establis h a kind o f relative equipollenc e i n means o f expression betwee n it an d th e elementar y theor y o f relations. Namely , i f we assume w e have a pai r of conjugate d quasiprojections , the n fo r an y elementar y statemen t X w e ca n effectively construc t a n equatio n X* i n th e calculu s tha t i s equivalen t t o X. This i s a preliminary an d muc h weake r for m o f (iii ) above ; i t doe s not concer n itself wit h th e proble m o f equipollence i n means o f proof. Wit h it s help, Tarsk i proved that an y decisio n procedure fo r th e se t o f true equation s i n the calculu s of relations would bring with it a decision procedure fo r the elementary theory o f relations, i n contradiction t o a result o f Church [1936 ] and Kalmar [1936] ; hence the se t o f true equations i n the calculu s o f relations i s not recursiv e (se e 8.5(xii ) below). Thi s theore m wa s announce d i n Tarsk i [1941] , p. 88 . Lemma s I-II I o f the abstrac t Tarsk i [1953 ] give a rough outlin e o f Tarski's origina l proof . Tarski [1941 ] presented an interesting formalization o f the calculus of relations as a deductive discipline. Th e language contained (binary ) relation variables, but no individua l variable s o r quantifiers , an d althoug h sententia l connective s wer e present, i t wa s pointed ou t o n p. 8 7 o f op. cit. that a n equivalen t formalizatio n involving onl y equations , i.e. , withou t sententia l connectives , coul d b e given . (Such a formalization wa s essentially carrie d ou t i n Chin-Tarski [1951]. ) Tarsk i proposed a finite se t o f axioms fo r the calculu s (essentiall y equivalen t t o the se t of axiom s give n i n Chin-Tarsk i [1951] , a s note d i n ibid., p. 352 , footnot e 10) , indicated tha t h e coul d deriv e al l th e hundred s o f law s occurrin g i n Schrode r [1895] on the basi s o f these axioms , an d aske d whethe r ever y tru e la w (tru e fo r all domains o f individuals) i n the calculus was so derivable. A s mentioned above , this proble m wa s subsequentl y answere d negativel y b y Lyndo n [1950] , and , i n fact, Mon k [1964 ] showe d tha t th e se t o f tru e equation s o f the calculu s i s no t finitely axiomatizabl e a t all . Nevertheless, just a s in the cas e o f expressibility, Tarsk i wa s able to establis h a kin d o f relative equipollenc e i n mean s o f proof betwee n hi s axiomatizatio n o f the calculus and the elementary theor y o f relations. Thi s i s essentially the resul t stated abov e in (iii ) (whe n reformulated fo r the calculus o f relations). Fro m thi s Tarski conclude d tha t th e se t o f equations derivabl e fro m hi s se t o f axiom s fo r the calculus o f relations is not recursive (se e 8.5(xii)). Further , sinc e his theorem PREFACE xvn reduced ever y proble m concernin g the derivabilit y o f a mathematical statemen t from a set o f axioms to the problem o f whether an equation i s derivable from a set of equations i n the calculu s o f relations, i n principle the whol e o f mathematica l research coul d b e carrie d ou t withi n th e framewor k o f thi s calculus . Thes e theorems wer e obtaine d b y Tarsk i durin g th e perio d 1943-1944 , an d presente d for th e firs t tim e i n hi s semina r o n relatio n algebra s a t th e Universit y o f Cali - fornia, Berkeley , durin g th e year 1945 . Reference s t o thes e theorems , a s wel l a s to th e Berkele y seminar , ca n b e foun d i n Chin-Tarsk i [1951] , pp. 341-343 ; se e also Chi n [1948] , pp. 2-3. Th e abstract s Tarsk i [1953] , [1953a] , [1953b] , [1954] , [1954a] contain announcements o f these theorems and several o f the other result s (also dating fro m th e 1943-194 4 period) that wer e referred t o i n the firs t par t o f this foreword . Roughly speaking , the formalis m £ x tha t i s the centra l focu s o f this wor k i s obtained fro m Tarski' s equationa l formalizatio n o f th e calculu s o f relation s b y introducing th e constan t E an d deletin g al l variables.

Tarski's formalizatio n o f set theor y i n £x wa s certainly no t th e firs t attemp t to eliminat e th e us e o f variables i n formalizin g mathematics . Probabl y th e ear - liest result s i n thi s directio n appeare d i n Schonfinke l [1924] . There , a kin d o f calculus o f unar y function s wa s developed . Basically , Schonfinke l considere d three distinguishe d unar y function s (late r calle d combinator s b y Curry) , C , S , and [/ , an d on e binary operatio n o n an d t o unary functions : tha t o f applying a unary functio n / t o a n argument a: , the resul t bein g represented b y juxtaposing the two, as in "fx n. Th e definitions o f C, S, and U are not simple. Eac h o f them has the property that, whe n applied to a unary function , i t yields another unar y function; thu s eac h o f the m take s o n unar y function s a s bot h argument s an d values. Fo r example, C i s the functio n that , whe n applied to any unary functio n /, yield s a constant unar y function , an d i n fact th e functio n constantl y equa l t o /, i.e. , Cf i s the unar y functio n which , when applie d t o an y argumen t x , yield s /, i n symbol s {Cf)x = /. Th e definition s o f S an d U ar e stil l mor e involved ; the reade r i s referred t o Schonnnkel' s paper . B y mean s o f the binar y operatio n of functional applicatio n w e can construc t furthe r unar y function s (calle d com - pound combinators ) fro m th e basi c three , fo r example , CC, C5 , (CS)C, etc . Schonfinkel indicate d ho w no t onl y boun d individua l variables , bu t als o boun d variables o f higher order s can be eliminated fro m mathematica l statement s wit h the hel p o f combinators. Thu s th e expressiv e powe r o f his calculu s reache s fa r beyond the domai n o f first-order logic . Schonfinkel mad e no attempt t o set up a deductive apparatus fo r his calculus. This tas k wa s take n u p b y Curr y an d hi s collaborators , startin g i n th e lat e 1920s, and prove d t o b e quit e involved . W e shall no t attemp t t o describ e thei r many achievements—th e reade r i s referre d t o book s o n combinator y logi c (a s this domai n i s no w called) , an d i n particular t o Curry-Fey s [1958 ] and Curry - Hindley-Seldin [1972] , which contai n extensiv e bibliographies . Rather , w e shall XV111 PREFACE briefly contrast the character o f the results presented in this book with those that have bee n obtaine d i n th e domai n o f combinatory logic ; moreover , w e contras t them onl y a s regard s th e specifi c proble m o f developin g part s o f mathematic s within a variable-free formalism . First o f all , i n contrast t o th e expressiv e power s o f combinator y logic , thos e of £x d o no t overreac h first-order logic , an d (a s wa s pointe d ou t above ) actu - ally compris e bu t a wea k par t o f it , namel y th e first-order logi c o f thre e vari - ables. Onl y unde r certai n additiona l assumption s (satisfie d b y mos t system s o f set theor y an d variou s systems o f arithmetic) doe s £x becom e equipollent wit h first-order logi c i n means o f expression. Simila r remark s appl y t o the deductiv e powers o f £x. Secondly , th e metho d presente d i n thi s boo k fo r formalizin g a given first-order syste m withi n £ x i s quit e general ; i t ca n b e applie d almos t mechanically t o man y differen t mathematica l theories . I n contras t t o this , th e various attempt s t o develo p differen t part s o f mathematics withi n combinator y logic hav e bee n quit e specifi c i n character , an d th e approache s use d hav e de - pended o n the particular theor y to be formalized. Finally , each o f our correlate d systems i n £x i s show n t o b e equipollen t wit h th e origina l first-order syste m in a stron g an d precisel y define d sens e tha t entails , e.g. , th e equiconsistency , equicompleteness, an d equidecidabilit y o f the tw o systems . I t i s not a t al l clea r to what exten t variou s first-order system s an d thei r combinator y analogue s ar e equipollent. Indee d th e ver y problem o f the consistency , o r relative consistency , of systems formalized i n combinatory logi c has traditionally pose d difficulties ; i n several case s th e answe r prove d t o b e negativ e an d som e o f these problem s ar e still open . In the late 1940 s and earl y 1950 s there began som e work which has a bearin g on the problem o f formalizing mathematic s withou t variables . Variou s algebrai c theories wer e developed tha t ar e analogue s o f first-order logic , much a s Boolea n algebra i s an algebrai c analogu e o f the sententia l calculus . Th e creatio n o f th e theories o f relatio n algebra s b y Tarski , an d o f projectiv e algebra s b y Everett - Ulam [1946 ] may be viewed a s preliminary step s i n this direction. Th e theory o f cylindric algebras , perhaps th e most extensivel y develope d o f such theories, was created b y Tarsk i i n collaboratio n wit h hi s forme r student s Louis e Chi n (Lim ) and Frederic k Thompso n durin g th e perio d 1948-52 , an d furthe r develope d b y Tarski, Henkin , an d Monk . A detaile d presentatio n o f various portion s o f thi s theory ca n b e foun d i n Henkin-Monk-Tarski [1971] , [1985] . Th e closel y relate d theory o f polyadic algebras was created by Halmos in the mid 1950s ; the relevant papers can be found i n Halmos [1962] . Othe r noteworthy theories of this type can be foun d i n Bernays [1959 ] and Crai g [1974] . Th e specifi c proble m o f using suc h algebraic theorie s t o construc t formalism s whic h contai n n o (bound ) variables , quantifiers, o r sentential connectives , and which are equipollent i n some sense to first-order logic , is addressed i n these last tw o works and i n Quine [1960] , [1971] . An excellen t discussio n o f variou s approache s t o thi s proble m ca n b e foun d i n Quine [1971] . PREFACE xix

With th e exceptio n o f som e algebrai c notion s an d result s i n Chapte r 8 , thi s work i s intende d t o b e largel y self-contained , an d accessibl e no t onl y t o mathematicians an d logicians, but als o to computer scientists , philosophers, an d others wh o may b e interested i n foundational research . In §§1.2-1.5 , 2.1-2.3, and 3.1-3. 4 w e respectively describ e th e formalism s £ , £+, an d £ x, an d thei r interrelationship . Afte r readin g thes e portion s o f th e book, it i s possible to proceed directly to the main equipollence results presente d in §§4.1-4. 5 and 7.1-7.2 , omittin g th e intervenin g text . §§4. 6 and 7.5-7.7 , con - cerning the formalizability o f various systems o f set theory and arithmetic in £ x, essentially depen d onl y o n §§4.1-4. 5 an d 7.1-7.2 , respectively . A more detaile d picture o f th e interdependenc e o f differen t section s o f the boo k i s presented i n the diagram s followin g th e table o f contents. Alfred Tarsk i Steven Givan t Berkeley, Californi a October, 198 3

Postscript

Alfred Tarsk i die d o n October 27 , 1983 , shortly afte r th e manuscript fo r thi s work wa s completed. Wit h hi s passing , th e worl d ha s los t a grea t logicia n an d an inspirin g teacher , an d I hav e los t a loya l friend . I n th e perio d sinc e hi s death i t ha s becom e apparen t tha t certai n smal l addition s shoul d b e mad e t o the text . Fo r example , i n th e las t fe w year s a numbe r o f interestin g result s have bee n obtaine d tha t hav e a direc t bearin g o n som e o f th e ope n problem s stated here . I n particular , afte r receivin g preliminar y copie s o f the manuscript , Roger Maddux, Hajna l Andreka , an d Istvan Nemet i solve d several o f these ope n problems. I n addition, variou s relevant results i n the literature that appeare d i n the lat e 1970 s and earl y 1980s , an d wer e overlooke d b y us, have bee n calle d t o my attention b y Andreka , Maddux , an d Nemet i a s well a s by Georg e McNulty . Rather tha n amendin g th e tex t itsel f a t thi s point , I hav e decide d t o includ e some additional footnotes , indicate d b y a n asteris k (*) , to discus s these results . Steven Givan t Berkeley, Californi a January, 198 6 This page intentionally left blank PREFACE xxi

Acknowledgments

It i s a grea t pleasur e t o acknowledg e ou r indebtednes s an d t o expres s ou r gratitude t o thos e wh o helpe d u s a t variou s stage s durin g th e preparatio n o f this book . Bot h Professo r Georg e McNulty , fo r a perio d o f severa l months , and Professo r Roge r Maddux , fo r almos t a year , assiste d Tarsk i i n hi s wor k on certai n portion s o f th e manuscript . I n addition , the y rea d ove r th e final draft an d mad e man y valuabl e suggestions . Professo r Joh n Corcora n als o rea d through th e final draf t i n a ver y carefu l an d thoroug h way . H e pointe d ou t several technica l inaccuracie s an d mad e innumerabl e detaile d suggestion s a s t o how th e styl e an d expositio n migh t b e improved . Drs . Hajna l Andrek a an d Istvan Nemeti carefully checke d many proofs i n the final draft. Professo r Rober t Vaught reviewe d preliminar y version s o f th e introductio n an d helpe d u s ver y much to improve its presentation. Th e author, subject , an d symbo l indices were meticulously prepare d b y Mr. Pete r Baker . Professo r Leo n Henkin helped u s to settle man y practica l problem s tha t aros e i n connectio n wit h ou r wor k o n thi s book. We als o wis h t o acknowledg e th e suppor t w e receive d fro m th e Nationa l Science Foundatio n throug h grant s t o th e Universit y o f Californi a a t Berkele y (Grant Nos . GP-27920 , GP-35844X , MCS74-223878 , and MCS77-22913) . This page intentionally left blank Bibliography

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Set-theoretical notion s x E A a : is a member o f A, 2 x $L A £ is not a member o f A, 2 {x: X[x]} clas s o f elements x satisfyin g X , 2 {t: X} clas s o f elements t[x] such that x satisfie s X, 2 0 empt y set , 2 {x} se t whos e onl y member i s x, 2 ; se e a/s o 21 8 {x, y} se t whos e onl y members ar e x an d y , 2 ; se e also 218 ACB,5Di A i s included i n B, 2 A C B, B D A A is properly include d i n 5, 2 ; se e a/s o 21 8 56A clas s o f all subsets o f A, 2 AU5 unio n o f y l and B, 2 AflB intersectio n o f A an d JB , 2 (J C unio n o f all members o f C, 2 p|C intersectio n o f all members o f C, 2 p| 0 universa l class , universe o f discourse, 2 A~ B differenc e o f A an d B, 2 —I? complemen t o f B, 2 (x,y) ordere d pai r o f x an d ?/ , 2; se e a/s o 21 8 (x, ?/, z) ordere d tripl e o f x, y , and 2 , 2 /d identit y relation , 3 Di diversit y relation , 3 xRy (x,y)eR,3 R\S relativ e product o f R an d S , 3 R~x convers e o f i?, 3 DoR domai n o f i?, 3

283 284 INDEX O F SYMBOL S

RnR range o f R, 3 A]R domain restrictio n o f it ! to A , 3 R*A iZ-image o f A, 3 AxB Cartesian produc t o f A and £? , 3 Fx, F{x), F x, FW xth valu e o f F, 3 {Fniel) system {(i , i^): i £ 7 } indexed b y 7 , 3 th Fi 2 ter m o f {F z: z

£

{x0,.-.,xa-i) a-termed sequenc e fo r 0 < a < a;, 4 {x?:£

{x0,... ,x a-i} range o f (xo , •. •, xa-i) fo r 0 < a < CJ , 4 pR, pO rank o f i?, rank o f O, 4

O{xo,-..,xa-i) value Ox o f operation 0 a t x , 4

H=(U,Q) = {U,Q i)ieI algebraic structure , 1 5

*=(A,0) = (A,Oi) ieI algebra, 23 1 ra similarity typ e o f 21 , 231 ii={U,E) realization o f £, 1 1 n = (N,o,s,+,-) algebra o f natural numbers , 21 5 «H = (B , 0,1, <,+,-, N) structure o f real numbers, 22 6 a = absolutely fre e algebr a o f type (2,1,2,1,0) , 23 8 ~x ^ x ^ + ^ + congruence relation s o n ^3 , 238 and 24 0

IRRA clas s o f relation algebra s representabl e ove r infinite sets , 245 ORA clas s o f omega relatio n algebras , 254

Formalisms

£ formalis m o f the predicate logi c o f one binar y relation, 4 £+ formalis m o f the extende d predicat e logi c of one binary relation , 2 3 £ x formalis m o f the equationa l logi c o f one binar y relation, 4 5

£3 3-variabl e subformalis m o f £, 65 , 72 £3" 3-variabl e subformalis m o f £+, 65 , 72 £53 standardize d 3-variabl e subformalis m o f £, 8 9 £JS£ standardize d 3-variabl e subformalis m o f £+, 8 9 £(3) 9 1

£+3) 9 1

£n n- variable subformalis m o f £, 9 1 £+ n- variable subformalis m o f £+, 9 1 £ox subformalis m o f £x withou t th e associativ e la w for 0 , 8 9 £w x subformalis m o f £ x wit h wea k associativ e la w for 0 , 8 9 x x w £ versio n o f £ wit h + , ", replace d b y f 5 15 2 ££ versio n o f £x wit h +," , 0,w replace d b y | | and $ , 15 3 £x varian t o f ££, 15 8 ££ versio n o f £ x wit h +,",©, w, 1 replaced b y 0 , 154 £~ subformalis m o f £x withou t 1 , 15 5 £^ subformalis m o f £x withou t 1 , 15 7 £^ subformalis m o f ££ withou t 1 , 15 7 £JT subformalis m o f £x withou t 1 , 15 8 £x reduce d versio n o f £x, 15 8 £x commo n extensio n o f £x an d £x, 16 2

£^r reduce d versio n o f £^, 16 3

£^r reduce d versio n o f £^, 16 3

£JTr reduce d versio n o f £~, 163 T formalis m o f sentential logic , 165 286 INDEX O F SYMBOL S

M^ formalis m o f the predicat e logi c o f n + 1 binary relations, 19 1 JV[(n)+ formalis m o f the extende d predicat e logi c o f n + 1 binary relations , 19 1 M(n)x formalis m o f the equationa l logi c o f n+1 binar y relations, 19 1 y arbitrar y formalis m o f predicate logic , 1 4 7+ extende d versio n o f T , 20 5

?m m-variabl e subformalis m o f 3> , 209

?m+ varian t o f 3>m, 209 CPN formalis m o f elementary numbe r theory , 215 0>£ 3-variabl e subformalis m o f ?N, 221 ?R formalis m o f the elementar y theor y o f real numbers, 226 yA formalis m o f the first-order theor y o f RA's, 236 £A formalis m o f the equational theor y o f RA's, 251 £EA formalis m o f the equational theor y o f ORA's, 251 £ arbitrar y equationa l formalism , 232 ; also equationa l formalism o f type (2,2,1) , 259 £' equationa l formalism s o f type (2) , 259 £" commo n extensio n o f £ and £', 260

Systems

§ arbitrar y syste m formalize d i n £, 1 1 §+ syste m i n £+ correlate d wit h § , 3 0 Sx syste m i n £x correlate d wit h a Q-syste m § , 12 5 §3 syste m i n £3 correlate d wit h a Q-syste m 8 , 14 1 S3" syste m i n £3" correlated wit h a Q-syste m § , 141 Sx syste m i n £x correlate d wit h §x, 152 S~ syste m i n £T correlate d wit h §x, 155 §7 syste m i n £~ correlate d wit h §x, 15 7 §^ syste m i n L^ correlate d wit h S£ , 15 7 SJT syste m i n £J T correlated wit h §x, 158 Ss predicativ e versio n o f a system S of set theor y admitting prope r classes ^ 178 Su predicativ e versio n o f a system § o f set theor y excluding prope r classes , 187 Qi Quine' s syste m o f set theory i n New foundations ... , 13 4 INDEX O F SYMBOL S 287

Quine's syste m o f set theor y i n Mathematical logic, 134 Mostowski's syste m o f set theory , 135 ; also Morse's syste m o f set theory , 17 9 Ackermann's syste m o f set theory , 13 5 Bernays-Godel syste m o f set theory , 17 9 Zermelo's syste m o f set theory , 18 7 system o f elementary numbe r theory , 21 5 system o f Peano arithmetic , 22 2 extended Zermelo-lik e syste m o f the theor y o f finite sets , 22 3 proper Zermelo-lik e syste m o f the theor y o f finite sets , 22 5 system o f elementary rea l number theory , 22 6 recursively axiomatize d subsyste m o f ft, 22 6

from £+ t o £, 28 ; also from £3 " to £3, 75; also from M( n+1)+ t o M^n+1), 19 7 from £% to £x, 7 7 x from £^ 3) t o £ , 9 0 auxiliary mappin g o f 4> + int o $+, 107/ . from £ + t o £x, 10 9 and 12 2 auxiliary mappin g o f $+ int o II, 11 2 from £ t o £3, 14 2 bijective translatio n mappin g fro m £+ t o £ , 148; also from 7* t o IP * and fro m 7 t o ? , 206 from IP N to 0> N, 221 bijective translatio n mappin g fro m £+ t o £ x, 148; also from M( n+1)+ t o £+, 19 6 from M( n+2>+ t o £+, 19 9 from?toM(n+2), 20 4 fromiPN toM< 5), 22 1 fromM(n+2) to3\4 n+2),212 fromM^ t o M^, 22 1 from £ " t o £' , 26 0 288 INDEX O F SYMBOL S

Primitive an d defined symbol s o f formalism s

(vo, V\,..., Vk, •..) sequenc e o f variables, 5 x,y,z, etc . specia l variables , 5 + x 1 (first-order ) identit y symbo l i n £, £ , £ , £ 3, etc., 5 and 23 ; also sentential constan t i n T, 165 ; also individual constan t denotin g identity elemen t i n 3> A, £A, £ EA, 23 6 an d 251 E nonlogica l binar y predicat e i n £, £ +, etc. , 5 and 23 ; also sentential constan t i n T , 165 ; also individual constan t i n £ EA, 25 1 —• implication , 5 ; also weak implicatio n i n £ x, 159 -i negation , 5 V disjunction , 6 A conjunction , 6 «-• biconditional , 6 V universa l quantifier , 5 3 existentia l quantifier , 6

Vx0...xn-i compositio n o f universal quantifications , 6

3Xo...xn-i compositio n o f existential quantifications , 6 + absolut e additio n symbo l i n £+, £x, £3" , y A, £A, £ EA, 2 3 and 236 ; also addition symbo l in J> N and ? R, 21 5 and 226 ; also disjunction symbol i n 1, 16 5 absolute multiplicatio n symbo l i n £+, J* A, etc., 24 and 236 ; also multiplication symbo l i n 9 N and 9 R, 21 5 and 22 6 — complemen t symbo l i n £+, ? A, etc. , 2 3 and 236; also negation symbo l i n T , 16 5 © relativ e multiplicatio n symbo l i n £+, 3> A, etc., and i n £ , 23 , 236, and 259 ; also conjunctio n symbol i n T , 16 5

0 absolut e zer o predicate i n £+, £ x, £3 , 24 ; also individual constan t denotin g zer o i n ?N an d 9R, 21 5 and 226 ; also individual constan t denoting Boolea n zer o in 9A, etc. , 23 6 1 absolut e unit predicat e i n £+, etc. , 24 ; also individual constan t denotin g uni t i n 9N an d 1PR, 21 5 and 226 ; also individual constan t denoting Boolea n unit i n CP A, etc., 23 6 0 diversit y predicat e i n £+, etc. , 24 ; also individual constant denotin g the diversit y elemen t i n ?A, etc. , 23 6 =s (second-order ) identit y symbo l i n £+, etc. , 23; also (first-order) identit y symbo l i n CP N, TR, ?A, £ A, £ EA, £ , etc. , 21 5 and 23 2 < (second-order ) inclusio n symbo l i n £+, etc. , 25; also ordering relation symbo l i n CP R, 226; also (first-order) inclusio n symbo l i n CP A, etc., 23 6 -• implicatio n i n £ x, 5 2 -11 negatio n i n £ x, 5 2 t binar y operatio n symbo l o f £x an d £ , 151/. and 25 9 ||, $ binar y operatio n symbol s o f ££ an d £x, 15 3 and 15 8 0 binar y operatio n symbo l o f ££, 153/ . x =*• stron g implicatio n i n £ r , 15 9 <£=* stron g biconditional i n £x, 15 9 n F0,..., F n nonlogica l constan t o f M^ \ 19 1

Co,..., C n nonlogica l constant s o f ?, 20 2 S successo r operatio n symbo l i n T N an d T R, 21 5 and 22 6 N natura l numbe r predicat e i n T R, 22 6 A unary operatio n symbo l i n £ , 25 9 binary operatio n symbo l i n £ ;, 25 9

General syntactica l an d semantical notion s in x inde x o f x, 5 t(j>X se t o f variables occurrin g fre e i n X, 6 [X] closur e o f X, 6 290 INDEX O F SYMBOL S

X[xo/u>o,..., x n-i/un-i] resul t o f simultaneously substitutin g UQ, ..., u n-\

for xo,..., xn_i, respectively , i n X, 7 ; se e a/50 67

X[^0, • •., un-i] resul t o f simultaneously substitutin g -u 0,..., w n-i

for vo, . • •, vn-i, respectively , i n X, 7 ; se e a/50 67 Xe th e lef t sid e o f (equation ) X , 4 6 Xr th e righ t sid e o f X, 4 6 X1 X*-X r+X£-.Xr-,46 T se t o f variables o f £, 5

T3 {vo , vi, v2}, se t o f variables o f £3, 65

Tm {v 0,...,t?m-i}, 20 9 T/i, T/i[£] se t o f terms o f £ , 23 2 $, $[£] se t o f formulas o f £, 6 ; se e a/5 0 23 2 and 25 9 • [? ] se t o f formulas o f (P , 15 *[S] se t o f formulas o f §, 1 8 $+ se t o f formulas o f £+, 2 5 $3 se t o f formulas o f £3, 6 5 $3" se t o f formulas o f £3", 75

$(3) se t o f formulas o f £ fo r whic h ever y subformul a has at mos t thre e fre e variables , 9 0 $is se t o f formulas o f £ + fo r whic h ever y subformul a has at mos t thre e fre e variables , 9 0

$n se t o f formulas o f £n, 9 1 $+ se t o f formulas o f £ J, 9 1 <&A se t o f formulas (equations ) o f £ A, 25 1 $EA se t o f formulas o f £ EA, 25 2 $' se t o f formulas o f £' , 25 9 $" se t o f formulas o f £", 259/ . E, E[£] se t o f sentences o f £, 7 ; se e also 233 E[5] se t o f sentences o f J, 1 6 E[S] se t o f sentences o f § , 1 9 E+ se t o f sentences o f £+, 2 5 Ex se t o f sentences o f £x, 4 5 E3 se t o f sentences o f £3, 65 E3" se t o f sentences o f £3", 75

E(3) »(3 ) n E, 9 0 E+3) *+ 3)ns+,90

En se t o f sentences o f £n, 9 1 E+ se t o f sentences o f £+, 9 1 INDEX O F SYMBOL S 291

x S rx set o f predicate-sentences o f £ , 15 8 set o f sentences o f £x, 16 2 S A set o f sentences o f £ A, 25 1 EA SEA set o f sentences o f £ , 25 2 n,n[£+] set o f predicates o f £+, 23 ; see also 45 ir set o f predicates o f £T, 15 6 A,A[£] set o f logical axiom s o f £, 8 A+ set o f logical axiom s o f £+, 2 5 Ax set o f logical axiom s o f £ x, 4 6

A3 set o f logical axiom s o f £3, 65 and 7 2 + set o f logical axiom s o f £3", 72 A3 A?' alternate se t o f logical axiom s o f £j, 6 9 set o f logical axiom s o f £n, 9 1 A+ set o f logical axiom s o f £+, 9 1 A~ set o f logical axiom s o f £^, 15 5 x x Ar set o f logical axiom s o f £ , 160/ .

A[?m] set o f logical axiom s o f 9m, 20 9 Ae,Ae[§] set o f nonlogical axiom s o f §, 1 1 A*+ set o f nonlogical axiom s o f S+, 3 0 A** set o f nonlogical axiom s o f Sx, 12 5

A$3 set o f nonlogical axiom s o f §3, 14 1 AT set o f nonlogical axiom s o f 8^, 15 5 h,h[£ ] relation o f derivability i n £, 8 ; see also 30 and 259 h[5] relation o f derivabilit y n?, 1 6 h[8] relation o f derivabilit y n 8 , 1 9 h+ relation o f derivabilit y n £+ , 2 6 hx relation o f derivabilit y n £x, 4 6

H 3 relation o f derivabilit y n £3, 65 and 7 2 relation o f derivabilit y n £j, 7 5 X x Hr relation o f derivabilit y n£ , 16 0 relation o f derivabilit y n £ x, 16 2

relation o f derivabilit y n 3> m+, 20 9 l~m+l relation o f derivabilit y n?m+i,209 H relation o f derivabilit y n £' , 25 9 h" relation o f derivabilit y n £" , 259/ . relation o f logical equivalenc e i n £, 9 ; also relation o f semantical equivalence , 1 3 relation o f logical equivalenc e i n 3 ^ 20 = + relation o f logical equivalenc e i n £+, 2 6 — X relation o f logical equivalenc e i n £ x, 4 9 292 INDEX O F SYMBOL S

=3 relation o f logical equivalenc e i n £3, 65 and 7 4 _+ =3 relation o f logical equivalenc e i n £3" , 76 #hX X i s derivable fro m # (i n £), 8 ; se e a/5 0 30 $ h X fo r ever y X G 0, 9 ; se e a/s o 3 0 Y\-X {7} h X, 8 ; se e also 30 \-x 0 h X, X i s logically provable , 9 ; se e a/s o 3 0

\J> and Q are equivalent o n the basis o f $ (i n £), 9 # = n \I> =0 Q , \I > and Q are logicall y equivalent , 9 ; also $ an d Q are semanticall y equivalent , 1 3 X=s> Y $ h [X <- • F), 1 0 0r/# theory generate d b y \I > in £, 9 ; also theory generated b y $ i n £ , 23 3 and 25 9 0^y theory generate d b y {Y} (i n L o r £) , 8/ . 0if*[3] theory generate d b y \ £ in !F , 20 6i|*[S] theory generate d b y ^ i n 8 , 2 0 0r/+# theory generate d b y $ i n £+, 2 6 0rjx# theory generate d b y # i n £x, 4 7

0r/3tf theory generate d b y ^ i n £3, 8 8 0*/+* theory generate d b y \I > in £3" , 88 ei?n* theory generate d b y ^ i n £n, 9 2 07/+* theory generate d b y $ i n £+, 9 3 x ©?/*# theory generate d b y \I > in £ r , 16 7 0r/A# theory generate d b y \I > in £ A, 25 1 0r/EA# theory generate d b y # i n £ EA, 25 2 0r/'# theory generate d b y $ i n £' , 25 9 0r/"# theory generate d b y $ i n £" , 259/ . KN[£] relation o f consequence i n £, 12 ; se e a/s o 2 7 and 4 7 $NI X i s a consequence o f ^ (i n £), 1 2 ^NX[J] X i s a consequence o f ^ i n J, 1 2 t=X 01 = X, X i s logically valid , 1 3 0pil theory o f il, 1 2 0pK theory o f K , 1 2 Den denotation function , 17 0 RE,RE[5] class o f realizations o f J, 1 6 MOX, M0X[5] class o f models o f X (i n IT) , 16 INDEX O F SYMBOLS

Special compoun d expressions

Bn 24 Ca 100 C(x,y,z) 179 G 195 G' 198

H0,..., H n 203, 221 H\,Hi,Hz 259 I 195 r 198 PAB — (PAB> PAB> • • •) 100/. Pn 110 P 129

PoiPli • • • iPm, • • • 204, 221 pV 196 QAB 96

Q{Ao,...,Am) 105 m, • • • 203, 220 i?0, • • • ,Rn+2 203 s 54 S' 54 5o,... ,53 63 S\,..., £ 5 180 S{ 184 Si, 62,53 188 Si, S3 188 &0, • • • , &n 204 Sx 174 T 55, 180

T\,..., T 6 131 rpl rpn rplll rpl rpl 11->1 12 > J5> J 6 133/.

Ti,..., T4 188

TQ, ... ,Tn 204 rpl rpl 10^' • ' ' 1 n 207 Ui,U2 128

UC 139 V(D 111 V,V',V" 154 v,va,vb 156/ XyZ 70 294 INDEX O F SYMBOL S

Xs relativization o f X t o the formul a Sx, 17 4 u x relativization o f X t o the formul a xEu, 18 7

Special sets o f sentences r,r,r 264 185 A e 185, 18 8 e',e" 260 251 So 252 £ 180, 18 8 £' 184 $ 187 187 189 #0 252 n 139, 17 8 178

Numeration o f schemata

(AI),... •, (AIX) axiom schemat a o f £ an d £+, 8 (FI),... ,(FV) five properties o f syntactical formalisms , 1 7 (DI),... .,(DV) axiom schemat a o f £+, 2 5 (BI),... ,(BX) axiom schemat a o f £ x, 4 6 (R) schema o f equivalent replacement , 6 5 (AVI') general schem a o f simple substitution, 6 6 (AIX') general Leibni z law , 6 8 (AIX") transposition schem a (varian t o f (AIX')) , 7 0 (AX) general associativit y schema , 6 8 (AX') variant o f (AX) , 7 0 (DI'),.. • ,(DIV) variants o f (DI),..., (DIV), 6 9 (BIV) weakened associativ e law , 8 9 (I),..., (IV) schemata definin g f i n terms o f +,~, w, an d conversely, 15 2 (V),... ,(X) schemata definin g || , 0, i n terms o f +,", ©, w, and thes e latter notion s i n terms o f || , 0, i , 153 INDEX O F SYMBOL S 295

(BF),..., (BX'), (I;) schemat a obtaine d fro m (BI),... , (BX), (I ) b y eliminating +,~, w o n the basi s o f (II),..., (IV), 152 (I), (I'), (I" ) variants o f (BVI) , 15 5 x (I),... ,(XVI) axiom schemata o f £r , 16 0 x (Ir),-- • , (XIIIr) improved axio m schemata o f £r , 160/ . (U),: -,(ivs) possible axio m schemat a o f £ *, 16 2 (C) impredicative comprehensio n schema , 17 7 (Cs) predicative comprehensio n schema , 17 8 (Cs') variant o f (C s), 18 9 (Z) Aussonderungsaxiom, 18 7 (Zu) predicative Aussonderungsaxiom , 18 7 (Zu') variant o f (Z u), 18 9 (PI),(PII),(PIII) axioms o f Peano arithmetic , 22 2 (In) induction schem a o f Peano arithmetic, 22 2 (QI),(QII),(QIII),(Is) arithmetic-like axiom s fo r th e extended Zermelo - like theory o f hereditarily finite sets , 22 3 (Cn) continuity schema , 22 6 (Ral), ..., (RaX ) axioms o f the theor y o f relation algebras , 23 5 This page intentionally left blank Index o f Names

Ackermann, W. , 135 , 141, 219, 248, Corcoran, J. , xxi , 28 1 273, 276 ; see also Ackermann's Couturat, L. , 164 , 27 4 system o f set theor y Craig, W. , xviii , 10 , 148 , 201, 274 Andreka, H. , xix, xxi, 40, 66, 71, Curry, H . B., xvii, 27 4 124, 141 , 143, 154, 209, 27 3

De Morgan, A. , xv; see also De Baker, P. , xxi Morgan's law s Bar-Hillel, Y., 27 8 Descartes, R. , see Cartesian Bernays, P., xviii, 130 , 179 , 185 , 186, 222, 229 , 273, 276; see also Dilworth, R . P. , 27 5 Bernays' system o f set theory , Bernays-Godel syste m o f set Ehrenfeucht, A. , 229 , 267, 275 theory Bernstein, F. , see Cantor-Bernstei n Everett, C, xviii , 27 5 theorem Birkhoff, G. , 48 , 233, 237, 243, 245, Fermat, P. , see Fermat's theore m 255, 27 4 Boole, G. , see Boolean Feys, R., xvii , 27 4 Borner, F., 153 , 154, 258, 274 Fraenkel, A., see Zermelo-Fraenke l system o f set theor y Frege, G., 165 , 275 Cantor, G. , see Cantor-Bernstein theorem Chin, L . H., xi, xvi, xvii, xviii, 48, Givant, S. , 35 , 54, 55, 70, 88, 89, 95, 138 , 268, 274 105, 113 , 144, 168 , 198 , 199, Chuaqui, R. , 178 , 179 , 274 206, 209, 210, 211, 226, 244, Church, A., xvi, 10 , 161, 166, 274 250, 256 , 267, 275 Cobham, A. , 28 2 Godel, K. , 13 , 130, 141, 179, 185, Comer, S . D., 154 , 27 3 186, 217 , 227, 275; see also

297 298 INDEX O F NAMES

Bernays-Godel syste m o f set Lyndon, R . C. , xi, xvi, 54 , 240, 27 7 theory Goldfarb, W . D. , 141 , 275 Maddux, R. , xix , xxi, 62 , 70, 89, 92, 93, 97 , 107 , 109, 138 , 143, 189, Hajek, P. , 130 , 282; see also 209, 244, 245, 268, 277 Vopenka-Hajek syste m o f set Mal'cev, A . L, 267 , 269, 270, 277 theory Markov, A . A., 268 , 269, 277 Hall, M., 269 , 270, 27 5 McKenzie, R . N. , xi, 54 , 55, 240, Halmos, P. R., xviii , 201, 275 259, 270 , 27 8 Henkin, L. , xviii, xxi, 1 , 2, 4, 12 , 16, McKinsey, J. C . C. , 6 8 24, 68, 71, 88, 141 , 201, 209, McNulty, G . F., xix, xxi, 55 , 234, 216, 231, 235, 238, 239, 242, 267, 269 , 270, 271, 278 245, 258 , 275 Mendelson, E. , 185 , 278 Heyting, A. , 27 4 Monk, J . D. , xvi, xviii, 1 , 2, 4, 12 , Hilbert, D., 222 , 229, 248, 276 16, 24, 62, 67, 88, 109 , 131, Hindley, R. , xvii , 27 4 141, 201 , 215, 216, 217, 218, Huntington, E . V., 50 , 276 231, 235 , 238, 239, 240, 242, 245, 258 , 268, 275, 278 Montague, R . M. , 15 , 128 , 187, 190, Jonsson, B. , 55, 237, 239, 240, 246, 276, 27 8 276 Morse, A., 1 , 130 , 131 , 164, 278; see also Morse's system o f set theory, Morse-Kelle y syste m o f Kalicki, J., 270 , 276 set theor y Kalish, D. , 15 , 276 Mortimer, M. , 141 , 278 Kalmar, L. , xvi, 10 , 276 Mostowski, A., 10 , 11, 135, 138 , 139, Kelley, J. L. , 1 , 131, 276; see also 175, 190 , 255, 257, 278, 281, Morse-Kelley syste m o f set see also Mostowski's syste m o f theory set theor y Korselt, A. , xi, xiii, xv, xvi, 54 , 61 Mycielski, J. , 22 9 Kuratowski, C, 12 9 Myhill, J . R. , 220 , 279 Kwatinetz, M . K., xvi, 63, 91, 92, 209, 210 , 276 Nagel, E., 28 2 Nemeti, I. , xix, xxi, 40, 66, 68, 71, Leibniz, G . W. , see Leibniz law , 90, 124 , 138 , 141, 143, 154, general Leibni z la w 209, 243, 273, 279 Levy, A., 189 , 190 , 27 6 Lewis, C. I., 164 , 27 6 Lindenbaum, A. , 57 , 256, 27 7 Peano, G. , see Peano arithmeti c Linial, S. , 168 , 277 Peirce, C . S. , xv, 279 ; see also Lowenheim, L. , xi, xv, 54 , 277 Peircean Lukasiewicz, J. , 270 , 27 7 Perkins, P., 269 , 270, 27 9 INDEX O F NAME S 299

Pigozzi, D., 264 , 27 9 Vopenka, P., 130 , 282; see also Post, E. L. , 168 , 268, 269, 270, 277, Vopenka-Hajek syste m o f set 279 theory Poznanski, E . I . J., 27 8

Wells, B. F., 27 7 Quine, W. V. O., xii, xiii, xviii, 8 , Woodger, J . H. , 28 1 65, 66 , 74, 75, 134 , 168 , 177, Wostner, IL , 63, 282 178, 279 ; see also Quine's system o f set theor y Yntema, M . K., 168 , 282

Rabin, M . O., 27 8 Robinson, A. , 27 8 Zermelo, E., 129 , 177 , 282; see also Zermelo's syste m o f set theory , Robinson, J . B. , 219 , 220, 28 0 Zermelo-Fraenkel syste m o f set Robinson, L . G., 27 4 theory, Zermelo-lik e theory o f Robinson, R . M. , 10 , 11, 138, 139, hereditarily finite set s 175, 255 , 257, 281

Schonfinkel, M. , xvii, 28 0 Schroder, E. , xv, xvi, 26 , 164 , 165, 280 Scott, D . S. , 141 , 186, 280 Schiitte, K. , 28 1 Seldin, J. P. , xvii, 27 4 Sheffer, H . M., see Scheffer's strok e Sierpinski, W. , 148 , 227, 280 Singletary, W . E. , 168 , 280 Skolem, T., 129 , 280 Suppes, P., 133 , 280, 28 1 Szmielew, W. , 138 , 280

Tarski, A., i , passim Thiele, E. J. , 190 , 281 Thompson, F. , xvii i

Ulam, S. , xviii, 27 5

Vaught, R. L., xxi, 128 , 138, 278, 282 von Wright , G . H. , 141 , 273, 282 This page intentionally left blank Index o f Subjects

The mai n referenc e t o a subject i s given in boldface type .

a-ary operation , 4 systems, 4 3 — relation, 4 — semantically equivalen t set s of a-termed sequence , 4 sentences, 5 4 absolute addition , 236 ; see also alphabetic varian t o f a formula, 7 4 absolute sum , Boolean additio n antecedent condition , 27 0 — implication; see strong arithmetic o f natural numbers ; see implication elementary numbe r theor y — multiplication, 24 , 236; see also — of real numbers; see elementary Boolean multiplicatio n theory o f real number s — product, xv , 25 associative law , 7 7 symbol; see absolute sum for relativ e products, 68 , 89/. , symbol 235, 243 — sum, xv , 25, 57; see also absolute associativity schema , 46 , 68/f., 89, addition, Boolea n additio n 160 symbol, xi/. , 23 atomic formula , xii , 24, passim — unit, xv ; see also Boolean uni t of a first-order formalism , 1 5 — zero, xv ; see also Boolean zer o of £, 5/. , 8 absolutely fre e algebra , 238 , 252 of £+, xii , 24/., 2 7 Ackermann's syste m o f set theory , — predicate, 23 , 45, 191^., 201, 135 205/., 237/. , 249 adjunction operation , 217 , 224 — term, 1 5 affirmation symbol , 16 5 Aussonderungsaxiom, 187 , 225 algebra, 231/f . automorphism, 5 9 algebraic logic , xviii/ . axiom o f choice, 96 , 128, 241 of one binary relation ; see — of constructibility, 18 6 equational logi c o f one binary — of extensionality, 129 , 131^., 135, relation 154, 185 , 22 4 — structure, 4 , 15, 16 almost identica l first-order — of infinity, 12 8 formalisms, 4 3 — of transitive embedding , 22 5

301 302 INDEX O F SUBJECT S

— of unordered pairs ; see pair axio m calculus o f relations, xvff. — schema, xi , passim; see also — of relatives; see calculus o f logical axio m schem a relations — set o f a system, 11 , 19/., canonical formulas , 79 / passim; see also logical axio m — sequence o f a formula, 6 , 12 , 110, of a theory, 9 , passim 112 of elementary numbe r theory , Cantor-Bernstein theorem , 14 8 215 cardinal; see cardinal numbe r of Boolean algebra , 5 0 — number, 4 of Peano arithmetic , 222^ . cardinality o f a set, 4 of the elementar y theor y o f real Cartesian powe r o f a class, 3 numbers, 22 6 — product o f classes, 3 of S+, 30 , 14 8 — space, 3 , 13 9 of Sx, 126# , 14 8 class o f singletons o f a class, 131/ ,

ofS3, 141 / 180 ofomega-RA, 254 / closure o f a formula, 6 / of RA,48 , 50 , 55, 235/, 25 1 combinator, xvi i axiomatic system , 2 0 combinatory logic , xvii / base o f a system, 30 , 36, passim; see common equipollen t extensio n o f two also axiom se t o f a syste m systems, 41/ , passim — of a theory; see axiom se t o f a commutative law , 7 7 theory compatible set s o f sentences, 9 , 52, Bernays-Godel syste m o f set theory , 139, 253/, 256/, 26 5 130, 133 , 135, 178/, 19 0 complement o f a class, 2 Bernays' syste m o f set theory , 130 , — of a binary relation , xv , 239 ; see 178/.; see also Bernays-Godel also complementation system o f set theor y — symbol, xi/ , 2 3 biconditional, 6 complementation, xi , xv, 25 , 57, binary predicate , xi , 5 , passim; see 236, 24 7 also predicate complete equationa l theory , 234 , — relation, xi , 3, 11/, passim 256#, 266 , 268 — representative o f a formula, 11 3 complete se t o f sentences, 9 , 1 3 boldface type , 2 — theory, xiii , 11 , 29, 35, 220; see Boolean addition , xi , 236 , 24 7 also complete equational theor y — algebra, xviii , 24 , 50/, 161 , 164/, completeness theorem ; see 235, 25 7 semantical completenes s — multiplication, 236 ; see also theorem absolute multiplicatio n composition o f existential an d — unit, 236 , 239 ; see also absolute universal quantifications , 6 unit — of functions, 3 — zero, 236 ; see also absolute zer o compound combinator , xvi i bound occurrenc e o f a variable, 6/ , comprehension schema , 177^". , 187, 232; see also law o f renamin g 190, 225 ; see also bound variable s Aussonderungsaxiom, INDEX OF SUBJECTS 303

impredicative comprehensio n cylindric algebra , xviii , 201 schema, predicativ e decidable theory , xiv , 10/, 30, 35, comprehension schema , 138, 167 ; see also decidable unrestricted comprehensio n equational theory , recursiv e schema theory, undecidabl e theor y conditional, 6 decidable equational theory ; see also — equation, 23 7 dually decidabl e equationa l congruence relation, 238/. , 240/., theory, essentiall y duall y 243, 252 decidable equationa l theor y conjugated projections , 96/ , 13 0 , 234 , 256/, 263/, 267, 26 9 — quasiprojections, xiii/ , xvi, 95/, decision problem ; see decidable 101, 106 , 172/, 192/, 200#, theory 227/ of the second degree , 257/, in a relation algebra , xiv , xvi, 265#, 269/ 242/, 248 # deduction theorem , 9/ , 26, 51$, 23 4 conjunction, 6 deductive power , 1 , passim; see also — symbol, 6 , 16 5 equipollence i n means o f proof consequence o f a set of sentences, of£+, 2 9 12/, 17 , 6 5 of £x, xviii , 53, 64, 66, 8 7

in a system, 1 3 of£3, 65 # consistent equationa l theory , 234, definable binar y relation , xiv/, 256#, 264# , 269 171^., 208 , 2i4/, 219/ — set o f sentences, 9 , 13 — relation, 171 , 21 8 — theory, xiii , xviii, 11, 29, 35; see — set, 173/ also consistent equationa l definitional extension , xii/ , 27, 37^., theory 42, 56, 62, 151#, 157 , 193 , constant symbol , 5 7 195/, 198 , 202/f., 212 , 223, 26 3 constructibility axiom ; see axiom of definitionally equivalen t structures , constructibility 216J7. continuity schema , 22 6 systems, xiv , 42/, 152#, 157 , continuum hypothesis , 16 8 193, 202, 206, 222, 225, 22 8 converse o f a binary relation , xv , 3, in the wider sense , 4 3 25; see also conversion theories; se e definitionally — symbol, xi/. , 2 3 equivalent systems , conversion, xi , xv, 52, 236, 24 7 polynomially equivalen t correlated syste m S+, 30 , 124,/f., varieties 138, 141, 148, 193^., passim DeMorgan's laws , 77 , 16 8 Sx, 126# , 134, 138, 141, 148 , denotable binar y relation , Ulff. 154/f., passim — set, 173/ S3, Ulff. denotation function , 170^* . x Sa , 152 , 15 7 — of a predicate, 26/ , 47, 56, 158/, x §b , 15 7 169#, 240 , 252# S~, 155/ — of a term, 253/ countable set, 4 denumerable set, 4 304 INDEX O F SUBJECT S derivability, xii , 7ff., passim — undecidable equationa l theory , — in a formalism, 16/f . 234, 254 , 256#, 264# , 267 ; see — in a first-order formalism , 1 5 also essentially duall y — in an equationa l formalism , 232/ . undecidable equationa l theor y — in elementary numbe r theory , 21 5 dyadic fraction , 22 7 — in £, xii , Sff. elementary numbe r theory , i, xiv/, on the basi s o f a set o f xviii/, 191 , 215#, 22 6 sentences; see derivability i n £ — theory o f natural numbers ; see relative to a set o f sentence s elementary numbe r theor y relative to a set o f sentences, — theory o f real numbers, xiv , 191, 9 226# — m£+, xii, 26/. — theory o f relations, xv i — in£x, xii, 46/, 25 2 elimination mapping ; see translation mapping — in £3, 65 — in £ x, 159J7. — of quantifiers, 80 , 109 , 113 , 229 x empty set , 2 , 128 , 133 ; see also — in £ s , 162 absolute zero , law o f the empt y — inO> +, 20 9 m set — in Peano arithmetic, 22 2 — relation, 4 — in sentential logic , 166 / equality symbol ; see identity symbo l derivative rul e o f inference; see equation, 25 , 45/, 232 , passim indirect rul e o f inference equational class ; see variety — universe, 5 7 — formalism, 48 , 232#, 251 , 259, difference o f two classes, 2 270/, passim direct alphabeti c varian t o f a — logic, 46/ ; see also equational formula, 7 4 formalism — product o f algebras, 244 / of one binary relation , 4 7 — rule o f inference, 8 — theory o f a class o f algebras, xiv , directly indecomposabl e algebra , 232, 23 3 237 of an algebra , 232 , 23 3 disjunction, 6 of relation algebras , xi , 23, — symbol, 6 , 161 , 165 251#, 26 8 distinguished elemen t o f an algebra , equipollence o f formalisms; see 231 equipollence o f system s distributive law , 51 , 77 — of £ an d £+, 27^". , passim diversity element , 23 6 in means o f expression , — relation, xv , 3 , 26 29 divisibility relation , 219 / in means o f proof, 2 9 domain o f a binary relation , 3 — of £3 and£+, 72 # — restriction o f a binary relation , 3 — of £x wit h £3, xi , xiii, 64#, 87 , dually decidabl e equational theory , 141 234, 254, 256#, 265# , 270 ; see — of £x wit h £j, 64# , 68 , 76#, also essentially duall y 107 decidable equationa l theor y — of £x with£ x, 152 / INDEX O F SUBJECT S 305

— of £x with£* , 15 3 of systems o f the theor y o f real — of £x with£ x, 160jf . numbers, 191 , 228/ x x x — of £ s wit h £ an d £ , 162 # of systems other tha n — of systems, xii/. , 30#, 40#, 126/ ; Q-systems, 229 / see also equipollent of Q-systems , 124# , 140 , 143/, formalizations o f systems, 147, 151 , 155/, 191#, 214 / equipollent formalization s o f of weak Q-systems , 202^". , Q-systems, relativ e 211J7- equipollence, stron g equivalence i n £ o n the basi s o f a set equipollence of sentences; see equivalence i n £ relativ e to a set o f sentence s in means o f expression, xii , xviii, 29 , 31^., 41, 58, 61, 136, relative to a set o f sentences, 151, passim; see also expressive 9/, 1 3 — relation, 111 , 239, 244/ power equivalent formulas , 10 , 1 7 in means o f proof, xii , xvi, 29, 31#, 41 , 58, 151, passim; — sets o f sentences, 9 , 1 1 essentially duall y decidabl e see also deductive powe r equational theory , 25 7 relatively t o a system, xiii , undecidable equationa l theory , 41/; see also relative 234, 254 , 257/, 264 , 267, 271 equipollence — £ x -expressible sentence , 136 / — of versions o f Morse's syste m o f — undecidable equationa l theory , set theory , 131 , 16 3 254, 256#, 263# , 269 / — results, xi , xiii/, 29^*. , 45, 53,/f., theory, xiv , 10 , 30, 64, 66, 68#, 72# , 87#, 152# , 35, 138# , 167 / 157, 160#, 165 , 167 ; see also existential quantification , 6 equipollent formalization s o f exclusion symbol ; see Sheffer' s systems, equipollen t stroke formalizations o f Q-systems , explicit definition , 218 / relative equipollenc e expressive power , 1 , 221, passim; — theorem, xiv , 30 ; see also first see also £ x -expressibility, equipollence theorem , secon d equipollence i n means o f equipollence theorem, prope r expression equipollence theore m of combinatory logic , xvii i equipollent extensio n o f a formalism ; of £,2 9 see equipollent extensio n o f a of £+,2 9 system of £x, 53J7- , 64, 79, 87, 90; of a system, xii/ , 30^. , passim; see also £ x -expressible see also equipollence o f system s sentence, equipollenc e result s — formalizations o f systems o f of £3, 88 , 90/; see also number theory , 191 , 216, £3-expressible sentenc e 220#, 22 5 of ££, 88,90 /

of systems o f set theory , 90 , of £n; see £ n-expressible 127#, 135 , 143 , 153#, 163 , 225 sentence, equipollenc e result s 306 INDEX O F SUBJECT S

of 0> m, 213/ . — logic; see formalism o f predicat e of?m+, 21 0 logic of the calculus o f relations, xv/ . formal language , 5 , 11, 18, 32, 163; extended predicat e logi c o f one see also formalis m binary relation , xii , 2 6 formalism, 16$". , passim; see also — Zermelo-like theory o f hereditaril y equational formalism , syste m finite sets , 225/ . — of first-order predicat e logic ; see extensional identit y relation , 2 5 formalism o f predicate logi c extension o f a formalism , 18 , passim — of predicate logic , xii , xiv, xviii, — of a system, xii , 20 , 226, passim 1, 4jf, 14# , 36#, 191# , 215 , extensionality axiom ; see axiom o f 229/, 231 , 234, 236, passim extensionality — with a deduction theorem , 51/ , — law, 224 ; see also axiom 139 formation o f complements; see of extensionalit y complementation Fermat's theorem , 16 8 — of converses; see conversion finitary operation , 4 formula, xii , 5ff., 15 , 18, 45, passim — part o f derivability, 1 0 — defining a relation, 17 1 finite set , 4 — of a first-order formalism , 1 5 finite schematizability , 6 2 — of an equational formalism , 23 2 finitely axiomatizabl e system , xiv , — of £, 5 / 138/, 177 , 189J7. — of £+, xii , 24/ of set theory , 167,179/ , — of £x, 4 5

1850 — of £3, 6 5 theory, xiv , 9 , 11 , 30, 35, 123, — of £n, 9 1 138, 268 # — of£+, 91 — based theory ; see finitely — of 3> m, 20 9 axiomatizable theor y — of 0> m+, 20 9 variety, 233 , 240, 245/, 254, — of the equationa l theor y o f 258, 263#, 267# , 27 1 relation algebras , 25 1 finitely generate d relatio n algebra , — without useles s quantifiers, 113 / U7ff. free algebra , 237# , 241 , 243, 252, finiteness axiom , 22 5 261/; see also absolutely fre e first axi s o f a Cartesian space , 3 , 13 9 algebra — equipollence theorem , 2 9 with definin g relations , 239 , first-order algebrai c structure ; see 243, 24 9 algebraic structur e — occurrence o f a variable, 6/ , — definitionally equivalen t passim structures; see definitionall y free generatin g se t o f an algebra , equivalent structure s 238/, 241J7 , 24 9 systems; se e definitionall y full relatio n algebr a o n a set, 239/f. , equivalent system s 244#, 250 , 256/ — formalism; see formalism o f function, 3 ; see also operation predicate logi c functional relation , 50 , 13 2 INDEX O F SUBJECTS 307

x fundamental component s o f a incompleteness o f £ , £3, £3 ; see formalism, 1 7 semantical incompletenes s o f — operations o f an algebra, 23 1 — relation o f a structure, 1 1 inconsistent se t o f sentences, 23 3 general Leibni z law , 68 , 92 of equations, 233/. , 266 — schema o f simple substitution, 66 , independence o f the associative law 72 for relativ e products, 6 8 generalized stron g translatio n index o f a variable, 5 mapping, 37/f. , 42 indirect rul e o f inference, 47 , 166 — translation mapping , 33/f. , 36 individual constant , 15 , 133, 232/ — union axiom , 22 5 individuals; see system o f set theor y generating se t o f a relation ring , admitting individuals , syste m 171, 200 , 214, 21 9 of set theory excludin g Greek type, 4 individuals groupoid, 258 ; see also theory of induction o n derivable sentences , 9 , groupoids passim hereditarily finite set , xiv , 217; see — on formulas, 9 , passim also extended Zermelo-lik e — on predicates, 2 4 theory o f hereditarily finite — schema, 222 , 225/f. sets, Zermelo-lik e theor y o f inequipollence o f £x wit h £ an d hereditarily finite set s £+, 45 , 53/f., passim — undecidable theory , 10/. , 30, 35 , in means of 123, 138#, 140, 23 4 expression, 53 / homomorphic image , 243^* . in means of homomorphism, 238# , 240/. , 243, proof, 54 / 249 inessential extensio n o f a theory, 25 5 Id-model, 5 8 infinite set , 4 identically satisfie d equation , 48 , interpreted formalism , 1 7 232, 237 , 240, 245#, 253 , 26 1 intersection o f a class, 2 identity element , 236 , 247 — of the empty set , 2 — function; see identity relatio n — of two binary relations ; see — relation, xi/. , xv, 3, 12, 246 absolute produc t — symbol, xi/. , 5, 124, 215, 232 , — of two classes, 2 passim invariance under a permutation, 5 7 iff, 6 / inverse o f a function, 3 IRRA, 245/ , 250, 255# implication symbol , 5/. , 9, 57 italic type, 5 , 15 impredicative comprehensio n £ x-expressible sentence , xvi , xviii, schema, 178 , 187/f. 62#, 90/f. , 136/f. , 176/; see inclusion relation betwee n classes , 2 also equipollence result s for £+, 2 5 £3-expressible sentence , xvi , xviii, for relatio n algebras , 23 6 62, 91/ incompatible set s o f sentences, 234 , £n-expressible sentence, 92 ; see also 254, 266 equipollence result s 308 INDEX O F SUBJECTS language; see formalism of a first-order formalism , 1 4 law o f adjoining a n element t o a set, of an equational formalism , 224 232 — of equivalent replacement , 66 , 72 of £, 5/ , 12 — of renaming boun d variables , 66 , of£+, 2 3 72, 74/., 9 2 of £x, 45,48 , 236

— of the empty set , 22 4 of£3, 65 , 69# x — of transitive closure , 22 5 of£a , 15 2 law o f syllogism, 16 8 of£x, 152 / Leibniz law , 68 ; see also general of£x, 16 2 x Leibniz law of£s , 16 2 lightface type , 2 ofM(n\ 19 1 logic o f a system; see theory o f a ofM^ + , 19 1 system of sentential logic , 165 / — of an equational formalism , 233 / logical equivalence , 9 , 13, 5 4 — of £, 9 ; see also predicate logi c of — object, xiii , 57 one binary relatio n — symbol, 56/f . — of£+; see extended predicat e logically complet e se t o f sentences; logic o f one binary relatio n see complete se t o f sentences — of £ x; see equational logi c o f one — consistent se t o f sentences; see binary relatio n consistent se t o f sentences — of£x, 16 6 — equivalent set s o f sentences; see logical axiom , 18 , passim logical equivalenc e of a first-order formalism , 1 5 — provable sentence , 9 , 13, 4 8 of £, 7 / — true sentence , 1 3 of £+, xii , 25/, 115# — valid sentence , 1 3 of £x, 46 , 152/ main mappin g theorem , 28 , passim

of £3, 6 5 for £ an d £+, 2 8

of£+, 69# , 84/ for £3 an d £3", 7 6 x of £n, 9 1 for £ and£+ , 8 7 of£+, 9 1 for £x an d £+, 110/f. , x of£a , 15 2 122/, 13 6 of£x, 15 3 for S and §3, 142 , 21 3 of /T, 155 / mapping; see function of£x, 159 / membership, 2

of 3> m, 20 9 — relation, xi , 14 of sentential logic , 16 6 — symbol, xi/ , 5, passim schema o f £, 7 / metalanguage, 1/ , 5 , 15 of £+, 25 , 115# metalogical constant , 5 of £x, 46^. , 50, 56, passim — operation, 6 of£x, 160# . — variable, 5 , 15 of£x, 16 2 metasystem, 1/ , 14 of sentential logic , 16 6 model o f a sentence, 12 , 16/f., 27, logical constant, 5/ , 57, passim 170, passim INDEX O F SUBJECT S 309

— of a set o f sentences, 12 , passim of the equationa l theor y o f — of a system, 13 ; see also representable relatio n algebras , realization o f a formalis m xvi — o f an equation , 23 3 number theory ; see elementary modus ponens, 8 , 52 , 92, 159 , 161, number theory , Pean o 166/, passim arithmetic, elementar y theor y Morse's system o f set theory , 1 , of real number s 130/, 163/ ; see also omega-relation algebra , 254/ , 258/, Morse-Kelley syste m o f set 264, 269/ theory one-one correspondenc e betwee n Morse-Kelley syste m o f set theory , classes; see one-one functio n 1, 131, 133, 135, 163/, 178/, — function, xv , 3 , 147jf. , passim 228 operation, 4 Mostowski's syste m o f set theory , — of detachment; see modus ponen s 135 — symbol, xi , 14/, 200 ; see also natural number , 4 ; see also operator elementary numbe r theor y operator, 23 , 56#, 151# , 15 8 negation, 6 ordered field , 22 6 — symbol, 5/ , 57 , 161 , 165 ordered pair , 2 , 132 , 18 0 nonnnitely base d variety , xiv , 240 , of numbers, 21 8 245, 26 8 — triple, 2 nonlogical constant, 5 , passim ordering relation, 218 / of a first-order formalism , 14/ , ordinal number , 3 191#, 21 0 pair; see ordered pair , unordere d of an equationa l formalism , pair 232 — axiom, 129/F. , 133 , 137, 154 , 176 , of £, 5 / 185, 189 , 197 , 225; see of £+, 2 3 also restricted pai r axio m of £x, 45,4 7 parentheses, 6 , 24 , 23 6 of elementary numbe r theory , partial orderin g relation , 6 3 215 Peano arithmetic , xiv , 19 , 191, ofomega-RA, 251,25 9 222#, 22 8 of Peano arithmetic , 22 2 Peircean addition ; see relative of RA , 25 1 addition of the elementar y theor y o f real — multiplication, 247 ; see also numbers, 22 6 relative multiplicatio n of the first-orde r theor y o f — unit; see identity elemen t relation algebras , 23 6 — zero; se e diversity elemen t of the equationa l theor y o f permutation, 57 , 72, 74, 82/, 25 0 relation algebras , 25 1 polyadic algebra , xvii i nonfinite axiomatizability , xiv , 179 , polynomially equivalen t varieties , 187, 19 0 258, 265 , 267 310 INDEX OF SUBJECTS possible definition , 25 , 37,/f., 154, for £+ an d £x, 12 3 156#, 195/. , 203/. , 207 , 223/., for § and S3, 143 226, 259 — inclusion, 2 , 21 8 schema, 25 , 40, 152#, 16 6 — relation algebra , 239/. , 246, 25 3 — realization o f a formalism; see on a set, 239/, 242, 244#, realization o f a formalis m 250, 254/; see also full relatio n postulates o f Boolean algebra ; see algebra on a set axiom set of Boolean algebr a provable sentenc e i n a system, 1 1 — of relation algebra ; see axiom set Q-relation algebra , xiv , 242^., ofRA 248#, 255 power o f a class; see Cartesian Q-structure, 172# , 192, 200, 208 , power o f a class 214, 216, 219/; see also weak — o f a set; see cardinality o f a set Q-structure — set axiom , 22 5 Q-system, xiii/ , 124#, 130, 132 , predicate, xii , 5, 23, passim 134/, 138 , 140/, 143#, 148 , — equation, 2 4 150, 172 , 191#, 200#, 205/, — logic o f one binary relation , 9 ; see 208^., 214#, 221jf., 229/, 249 ; also formalism o f predicate see also weak Q-syste m logic, logi c o f £ — in the narrower sense ; see — representing a formula, 11 3 Q-system predicate-sentence, 158 , 166, 17 0 — in the wider sense ; see weak predicative comprehensio n schema , Q-system 178, 187# quantifier, xi , xviii, 5/, passim — system o f set theory, xi v quantifier-free formula , 6 , 79, 23 7 admiting prope r — sentence, 25/ classes, 177 ff. quasiprojectional system ; see excluding prope r Q-system classes, 187$' . quasiprojections; see conjugated — version o f Zermelo's system o f set quasiprojections theory, 18 8 Quine's syste m o f logic, 6 5 product o f classes; see Cartesian — systems o f set theory, 134 , 17 7 product o f classes quotient algebra , 238/ , 240/, 243, projections; see conjugated 249/, 252/ projections range o f a binary relation , 3 projective algebra , xvii i rank o f a relation, 4 proof power ; see deductive powe r symbol, 1 5 proper classes ; see system o f set — of a set, 128 theory admittin g prope r — of a system o f set theory, 12 8 classes, system o f set theory — of an operation, 4 , 231 excluding proper classe s symbol, 15 , 23 1 — equipollence theorem s fo r £> — of an operator, 2 3 and £+, 27 , 2 9 rationally equivalen t varieties ; see

for £3 an d £3", 76 polynomially equivalen t for £x an d ££, 8 7 varieties INDEX O F SUBJECT S 311

real number; see elementary theor y — structure, 11 ; see also algebraic of real number s structure realization o f a formalism, 16/. , 56, relative addition , 24 , 236; see also passim relative su m — of a first-order formalism , 15 , 19 2 — equipollence, xiv , xvi, 95, 107, — of £ A, 25 1 123/., 14 7 — of £ EA, 251,25 3 of the calculu s o f relations an d — of £, 1 1 the elementar y theor y o f — of £+, 26 , 5 6 relations, xv i — of £x, 47 , 5 6 in means o f expression, xv i

— of ^3, 6 5 in means o f proof, xv i — of£x, 15 8 of £x wit h £, xiv , 95 , 123/, — of MM, 19 2 147/f., 242, passim — ofM^ + , 19 2 of £x wit h £+, xiv , 95, 107, — ofM(n)x, 19 2 123/, U7ff., 244 , passim — of sentential logic , 165/ . of £ an d £ 3, 14 2 — of the elementar y theor y o f rea l — implication; see weak implicatio n numbers, 22 6 — multiplication, 57 , 236; see also recursive, xvi , 10/. , 31, passim; see relative produc t also decidable theor y symbol; se e relative produc t — definition, 218/. ; see also symbol induction — product, xi , xv, 3 , 25, 18 0 recursively enumerable , 10/. , 21, 31 , symbol, xi/ , 2 3 passim — semantic completenes s o f £ x, 12 4 reduced versio n o f a formalism , — sum, x v USff. — unit; se e identity relatio n reduction method , 27 0 — zero; se e diversity relatio n reflexive relation , 5 0 relativization o f a formalism, 19 , 95 relation, 3 , passim; see also a-ary — of quantifiers, 174/. , 187,^. relation replacement o f equals by equals ; se e — algebra, xi , xiv, xviii, 48, 54, 62, rule o f replacement o f equal s 68, 231 , 235/f., 268#; see also by equal s axiom se t o f RA , free algebra , — of equivalent formula ; se e law o f full relatio n algebra , IRRA , equivalent replacement , schem a omega-relation algebra , prope r of equivalent replacemen t relation algebra , Q -relation — schema, 19 0 algebra, representabl e relatio n representable relatio n algebra , xiv , algebra, simpl e relatio n 54, 62, 239#, 250 , 255/, 26 8 algebra, SQR A representation proble m fo r relatio n — ring, 171 , 200, 214, 219, 23 9 algebras, xi , 5 4 — symbol, 5 , 14/ restricted extensionalit y axiom , — term, 2 3 134/, 15 4 relational se t algebra ; see proper — pair axiom , 63 , 131 , 176, 179, relation algebr a 186, 18 9 312 INDEX OF SUBJECTS

— singleton axiom , 131 , 135, 154 , of sentential logic , 16 6

186 of weak Q-system s i n Tm+ , restriction o f a binary relation ; see 210 domain restrictio n o f a binary theorem, 13/f. , 127 , 144, 210 ; relation see also semantical i?-image o f a class, 3 completeness rule o f inference, xi/. , 8, 47; see also — consequence; see consequence o f a direct rul e o f inference, indirec t set o f sentences rule o f inferenc e — equipollence i n means of for equationa l formalisms , expression, 31 , 151 166 — expansion o f a syntactica l for £, 8 formalism, 1 8 for £+, 2 6 — formalism; see interpreted x for£ , 4 7 formalism x for£ , 159 # — incompleteness o f £x, xi , 48, 55 , for sententia l logic , 16 6 123 — of replacement o f equals by of £3, £3", xi/ , 88 equals, xi , 47, 233 — notions i n formalisms, 11/f. , 16/f. , — of substitution, 233,25 2 26/, 31 , 47, 165/, 169 # satisfaction relation , 12 , 18, 26/, 47 , of a formalism, 16 / 112, 169 of an equational formalism , schema o f class construction; see 233 comprehension schem a of £, Uff. — of equivalent replacement , 66 , 183 of £+, xii , 26/ second axi s o f a Cartesian space , 3 of £x, 4 7 — equipollence theorem , 29/ , 136 of £ , 6 5 — identity symbol , 2 3 3 — soundness o f £ x, 5 3 second-order operatio n symbol ; see of£+, 5 4 operator semantical completeness , xiii , — equipollence i n means passim; see also relative of expression, 31, 151 semantic completenes s o f £x, semantically adequat e formalism , 1 7 semantical completenes s — complete formalism , 14/ , 17/, 27 , theorem 34; see also semantical of a first-order formalism , 15 , completeness theore m 207 system, 3 1 of an equational formalism , set o f sentences, 13/ 233, 243 , 251 — consistent se t of sentences, 1 3 of £, 13/ , 124, 12 7 — equivalent sentences , 17 , 150 , of £+, xii , 27, 101, 112, 124 , passim 127, 145 , 24 3 — incomplete formalism , xi/ , 27 ofM^)+, 19 3 sets o f sentences, 13 , 150 , of Q-systems , 127 , 24 1 passim

in £3, 14 2 — sound formalism , 17 , 53, 155^". INDEX O F SUBJECT S 313

system, 3 1 — axiom, 154 ; see also restricted semiassociative relatio n algebra , 24 3 singleton axio m semisimple algebra , 23 7 — of a number, 21 8 sentence o f a formalism, 16^". , 45, — part o f derivability, 10 , 33 ff. passim; see also formula SQRA, 244# , 25 0 — of £, xii , 6 standard mode l o f a set-theoretica l — of £+, xii , 2 5 system, 1 4 x — of £ , xii , 4 5 of elementary numbe r theory , — of £3l 6 5 215 — of £„, 9 1 of the elementary theor y o f real -of£+, 9 1 numbers, 22 6 — of? , 20 9 m — realization o f £, 1 4 — 3> , 20 9 m+ strong axio m o f infinity, 12 8 — of the equational theor y o f — equipollence o f systems, 37^". , 42 relation algebras , 25 1 in means o f expression , sentential connective , xi , xvi, xviii, 5, 45, 16 5 37/ — constant, 1 5 — implication, 15 9 — logic, xiv , 161 , 164#, 27 0 —0-structure; see Q-structure — tautology, 16 6 — subsystem, 37 , 3 8 sequence; see a-termed sequenc e — translation mapping , 37$*. , 43; — of conjugated quasiprojections , see also generalized stron g 101, 105/. , 20 2 translation mappin g — without repeatin g terms , 4 subalgebra, 240 , 244/f. , 25 0 set o f finite rank; see hereditarily subdirect produc t o f algebras, 237 , finite se t 240, 244/, 25 5 set-theoretical mode l o f a set o f subdirectly indecomposabl e algebra , sentences, 14 , 132 , 17 4 237, 242/, 25 5 of a system, 1 4 subformalism, 18 , 27, passim — realization o f £, 14 , 130/ . substitution o f variables, 7/ , 66/f. , — system; see system o f set theor y 72#, 92 , 125 , 232/, passim; set theory , xi , 1/. , passim; see also see also general schem a o f system o f set theor y simple substitution, la w o f shape o f symbols, 2 , 19 7 renaming bound variable s Sheffer's stroke , 15 2 in £ , 6 7 similarity typ e o f an algebra , 23 1 3 subsystem, 20 , passim simple algebra , 237 , 239, 245/. , subterm condition , 259 , 27 0 248#, 25 5 — substitution o f variables; see successor relation , 215 , 217,/f . substitution o f variables symmetric difference , 2 4 simultaneous substitution o f — division, 16 2 variables; see substitution o f — relation, 5 0 variables syntactical equipollenc e i n means o f singleton, 2 expression, 3 1 314 INDEX O F SUBJECTS

— formalism, 17 , 251 of set theory, extende d — notions o f a formalism, 16 / Zermelo-like theory o f of equational formalisms , 23 3 hereditarily finit e sets , in formalisms, 12/f. , 16/f. , 31; Morse-Kelley syste m o f set see also semantical theory, Morse' s syste m o f set completeness theore m theory, Mostowski' s system o f — part o f a formalism, 18 , 166 set theory , predicative syste m syntactically equivalen t sentences ; of set theory, Quine' s system s see equivalent formula s of set theory, Vop£nka-Haje k system, 18# , 30#, 41 jf., 151/. , system o f set theory, passim; see also formalism, Zermelo-Fraenkel syste m o f set system o f set theor y theory, Zermelo-lik e theor y o f — developed i n a formalism, 18/f. , hereditarily finite sets , passim; see also correlated Zermelo's syste m o f set theor y system admitting individuals , in an uninterpreted formalism , 128, 133#, 154/, 156/, 177 , 21 197 in £, 1 1 admitting prope r classes , in £+, 30 , passim 128, 130#, 133#, 154, 174, — formalized i n a formalism; see 177#, 19 0 system develope d i n a excluding individuals , formalism 128#, 134 , 154, 177, 22 8 — of arithmetic, xix , 215#, 222/f., excluding proper classes , 226jf.; see also elementar y 128#, 132jf. , 154 , 187jf., 189 number theory , — of the elementary theor y o f real elementary theor y numbers; see elementary theory o f real number s of real numbers , x Peano arithmeti c tautology o f £ , 46/ , 233/, 252, 26 0 — of T; see sentential tautolog y — of classes indexe d b y a class, 3 term, 15 , 232, 252/, 259/f. — of conjugated quasiprojections ; ith ter m o f an indexed system , 3 see sequence o f conjugate d £th ter m o f a sequence, 4 quasiprojections theory, xiii , 9, passim; see also — of elementary numbe r theory ; see complete theory , consisten t elementary numbe r theor y theory, decidabl e theory , — of Peano arithmetic ; see Peano system, equationa l theory , arithmetic undecidable theor y — of set theory, xii , xiv, xix, 1 , 11 , — based upo n a set o f sentences; see 14/, 45 , 95, 124, 127#, 143 , theory generate d b y a set of 147, 153/r. , 163 , 170, 177#, sentences 186, 187/, 189#, 224/f.; see — generated b y a set o f sentences, also Ackermann's syste m o f set xi, 9 theory, Bernays-Gode l syste m of equations, 23 3 of set theory, Bernays ' syste m — in a system, 1 1 INDEX O F SUBJECT S 315

— of a class o f structures, 12 ; see — in sentential logic , 165 / also equational theor y o f a ultrafilter, 24 5 class o f algebra s ultraproduct, 24 5 — of a structure, 12 ; see also undecidable equationa l theory , 255 , equational theor y o f an algebr a 257/, 266 # — of a system, 11 , 20, passim — theory, xiv , 10/, 63/, 123 , 138#, — of types, 5 7 140, 167/, 215/, 254; see also threefold implicatio n theorem ; see decidable theory , decidabl e deduction theore m equational theory , duall y three-variable formalisms , xi , 64/f . decidable equationa l theory , transitive number , 21 8 essentially duall y undecidabl e transitive relation , xv , 5 0 equational theory , essentiall y translation mapping , 28 , 32^., 36, undecidable theory , essentiall y 75#, 87#, 107J7. , 112#, 122/. , undecidable equationa l theory , 125#, 141/. , 147ff. 162/, 192 , hereditarily undecidabl e 197, 206/, 209 , 212#, 221/, theory, undecidabl e equationa l 249, 260 , 265, passim ; see also theory equipollence results , undecidability o f the equationa l equipollent formalizations , theory o f relation algebras , generalized stron g translatio n xiv, 25 5 mapping, generalize d of representabl e relatio n translation mapping , stron g algebras, xvi, 25 5 translation mappin g uninterpreted formalism ; see from £ + t o £, 28/f. , 125 , syntactical formalis m 147#, 197 , 206, 209, 24 9 union axiom , 63 , 131, 135, 180 , 18 8

— -—from £3 " to £ 3 , 75 / — of a class , 2 from £3 " to £ x 77J7. — of two binary relations ; see from £ + t o £x, 107/f. , 147# , absolute su m 197 — of two classes, 2 from L to£x, 141 / universal class , 2 , 19 0 from £ x to£x, 162 / — quantification, 6 -from £ x to£x, 16 2 — quantifier, 5 , 57, 175 , 232 transposition schema , 7 0 — relation fo r finite sets , 216 / true sentenc e i n a system, 1 3 — relation fo r two-elemen t sets , 139 , of a structure, 1 2 172/, 201 , 219 true equationa l sentenc e i n a n universe o f a realization o f £, 1 1 algebra, 25 3 — of an algebrai c structure , 1 5 truth i n elementary numbe r theory , — of an algebra , 23 1 227 — of discourse, xv , 1 4 — in£, 1 2 universally quantifie d equation , 23 2 — in£+, 2 7 — valid sentence , 24 8 — in£\ xiv , 47 , 17 0 unordered pair , 2 ; see also pair — in£x, 158 / axiom, restricte d pai r axio m — in£x, 16 2 of numbers, 21 8 316 INDEX O F SUBJECT S unrestricted comprehensio n schema , 177/ valid sentence , xi , 1 3 xth valu e o f a function, 3 variable, xi , xvii, 5 , passim; see also bound occurrenc e o f a variable, free occurrenc e o f a variable, general schem a o f simple substitution, la w o f renamin g bound variables , substitutio n of variable s — of a first-order formalism , 1 4 — of an equational formalism , 23 2

— of £3, 6 5

— of £n, 9 1 — of?m, 20 9 — of ?m+, 20 9 variety, 232/ , 235 , 240, 244#, 250# , — of groupoids, xiv , 258/, 264, 267# virtual theor y o f classes, xi i Vop£nka-Hajek syste m o f set theory , 130 weak implication , 15 9 weak 0-structure, 208 , 214/, 21 9 — Q-system, xiv , 200#, 206 , 208# , 227 well-foundedness axiom , 128 , 135, 185, 22 5 well ordered set , 3 , 5, 24 7 word problem, 268,/f . w.u.q. formula ; see formula withou t useless quantifier s Zermelo-Fraenkel syste m o f set theory, 1 , 128/, 133 , 135, 19 0 Zermelo-like theory o f hereditaril y finite sets , 225 ; see also extended Zermelo-lik e theor y of hereditarily finite set s Zermelo's syste m o f set theory , 18/ , 128/, 133 , 135, 177 , 187 , 189, 225, 22 8 Index of Numbered Item s

item page item page item page

1.2(i) 5 3.2(xxi)-(xxxiii) 50 3.9(v) 84 1.3(i)-(ii) 8 3.3(i)-(ii) 51 3.9(vi)-(ix) 87 1.3(iii) 9 3.3(iii)-(v) 52 3.9(x)-(xii) 88 1.3(iv)-(v) 10 3.3(vi) 53 3.9(BIV) 89 1.3(vi)-(viii) 11 3.4(i)-(iii) 53 3.10(i)-(ii) 90 1.4(i)-(vii) 13 3.4(iv)-(v) 54 3.10(iii)-(v) 91 1.6(FI)-(FV) 17 3.4(vi)-(vii) 55 3.10(vi) 92 1.6(i)-(iv) 19 3.5(i)-(ii) 57 4.1(i)-(v) 96 1.6(v) 20 3.5(iii)-(vi) 58 4.1(vi)-(viii) 97 2.1(i) 23 3.5(vii) 59 4.1(ix)-(x) 99 2.1(ii)-(iii) 24 3.5(viii) 61 4.1(xi)-(xii) 100 2.1(iv) 25 3.5(ix) 62 4.2(i)-(vi) 101 2.2(DI)-(DV) 25 3.6(i) 62 4.2(vii) 103 2.2(i)-(vi) 26 3.6(ii)-(iii) 63 4.2(viii) 104 2.3(i)-(ii) 27 3.7(i) 65 4.2(ix)-(xi) 105 2.3(iii)-(v) 28 3.7(AVI') 66 4.2(xii) 106 2.3(vi)-(xi) 29 3.7(AIX') 68 4.2(xiii) 107 2.4(i)-(ii) 31 3.7(AX) 68 4.3(i)-(v) 107 2.4(iii) 32 3.7(DIII') 69 4.3(vi) 108 2.4(iv)-(vi) 33 3.7(AIX") 70 4.3(vii)-(viii) 109 2.4(vii)-(ix) 35 3.7(AX') 70 4.3(ix) 110 2.4(x)-(xii) 37 3.8(i)-(iii) 72 4.4(i)-(iv) 110 2.4(xiii)-(xiv) 38 3.8(iv) 73 4.4(v)-(vi) 111 2.5(i)-(iv) 41 3.8(v)-(vi) 74 4.4(vii)-(xii) 112 2.5(v)-(vi) 42 3.8(vii)-(ix) 75 4.4(xiii)-(xv) 113 3.1(i)-(ii) 46 3.8(x)-(xii) 76 4.4(xvi)-(xvii) 114 3.1(BI)-(BX) 46 3.9(i) 76 4.4(xviii)-(xx) 115 3.1(iii) 46 3.9(H) 77 4.4(xxi)-(xxii) 116 3.1(iv) 47 3.9(iii) 79 4.4(xxiii) 117 3.2(i)-(xx) 49 3.9(iv) 80 4.4(xxiv)-(xxvi) 118

317 318 INDE X O F NUMBERED ITEM S

item page item page item page

4.4(xxvii)-(xxix) 119 5.4(viii) 164 7.5(vi) 220 4.4(xxx)-(xxxi) 121 6.1(i)-(iii) 170 7.6(PI)-(PIII) 222 4.4(xxxii)-(xxxiv) 122 6.2(i)-(iii) 171 7.6(In) 222 4.4(xxxv)-(xxxix) 123 6.2(iv)-(vii) 172 7.6(i) 223 4.4(xl)-(xli) 124 6.2(viii)-(xi) 173 7.6(QI)-(QIII) 223 4.5(i)-(iv) 125 6.3(i)-(ii) 174 7.6(Is) 223 4.5(v) 126 6.3(iii)-(v) 175 7.6(ii) 225 4.5(vi) 127 6.3(vi)-(viii) 176 7.6(Cn) 226 4.6(i)-(iii) 129 6.4(C) 177 8.1(i)-(iii) 232 4.6(iv) 130 6.4(i) 178 8.1(iv) 234 4.6(v) 131 6.4(CS) 178 8.2(i) 235 4.6(vi) 132 6.4(ii) 178 8.2(Ra I)-(Ra X) 235 4.7(i)-(ii) 136 6.4(iii) 179 8.2(ii)-(iii) 236 4.7(iii)-(vi) 138 6.4(iv) 180 8.2(iv)-(viii) 237 4.7(vii)-(ix) 139 6.4(v) 184 8.2(ix)-(x) 238 4.7(x)-(xi) 140 6.4(vi) 185 8.3(i)-(ii) 239 4.8(i)-(ii) 140 6.5(Z) 187 8.3(iii)-(vii) 240 4.8(iii)-(vi) 141 6.5(ZU) 187 8.3(viii) 241 4.8(vii)-(x) 142 6.5(i)-(ii) 187 8.4(i)-(iii) 242 4.8(xi)-(xiv) 143 6.5(iii) 188 8.4(iv)-(vi) 244 4.8(xv)-(xvi) 144 6.5(CS') 189 8.4(vii)-(x) 245 5.1(i) 148 6.5(iv) 189 8.4(xi) 246 5.1(ii) 150 6.5(v)-(vi) 190 8.4(xii)-(xiii) 248 5.2(I)-(IV) 152 7-1(0 192 8.4(xiv)-(xvi) 250 5.2(V)-(X) 153 7.1(H) 193 8.5(i)-(ii) 251 5.3(i)-(iii) 154 7.1(iii) 198 8.5(iii)-(iv) 252 5.3(iv) 155 7.1(iv) 199 8.5(v)-(vi) 253 5.3(1) 155 7.1(v) 200 8.5(vii)-(xi) 254 5.3(v) 155 7.2(i)-(ii) 201 8.5(xii) 255 5.3(I')-(I") 155 7.2(iii) 202 8.5(xiii)-(xv) 256 5.3(vi) 156 7.2(iv) 205 8.5(xvi) 257 5.4(i)-(iv) 159 7.2(v) 208 8.6(i)-(v) 260 5.4(XI)-(XVI) 160 7.3(i) 210 8.6(vi) 261

5.4(Ir)-(IXr) 160 7.3(ii) 211 8.6(vii)-(viii) 263 5.4(Xr)-(XIIIr) 161 7.3(iii) 213 8.6(ix)-(xi) 264 5.4(v)-(vi) 161 7.4(i) 214 8.6(xii)-(xiv) 265

5.4(IS)-(IVS) 162 7.5(i)-(iii) 216 5.4(vii) 163 7.5(iv)-(v) 217

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