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.