Addison / Henkin / Tarski The Theory of Models

Proceedings of the 1963 International Symposium at Berkeley
1. Auflage 2014
ISBN: 978-1-4832-7534-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 510 Seiten, Web PDF

Studies in Logic and the Foundations of Mathematics: The Theory of Models covers the proceedings of the International Symposium on the Theory of Models, held at the University of California, Berkeley on June 25 to July 11, 1963. The book focuses on works devoted to the foundations of mathematics, generally known as 'the theory of models.' The selection first discusses the method of alternating chains, semantic construction of Lewis's systems S4 and S5, and continuous model theory. Concerns include ordered model theory, 2-valued model theory, semantics, sequents, axiomatization, formulas, axiomatic approach to hierarchies, alternating chains, and difference hierarchies. The text also ponders on Boolean notions extended to higher dimensions, elementary theories with models without automorphisms, and applications of the notions of forcing and generic sets. The manuscript takes a look at a hypothesis concerning the extension of finite relations and its verification for certain special cases, theories of functors and models, model-theoretic methods in the study of elementary logic, and extensions of relational structures. The text also reviews relatively categorical and normal theories, algebraic theories, categories, and functors, denumerable models of theories with extra predicates, and non-standard models for fragments of number theory. The selection is highly recommended for mathematicians and researchers interested in the theory of models.

Autoren/Hrsg.

Addison, J. W.

Henkin, Leon

Tarski, Alfred

Inhaltsverzeichnis

1;Front Cover;1
2;The Theory of Models;6
3;Copyright Page;7
4;Table of Contents;8
5;PREFACE;10
6;EVERT WILLEM BETH;12
7;LIST OF PAPERS GROUPED BY SUBJECT MATTER;14
8;FOREWORD ON TERMINOLOGY;16
9;PART I: INVITED PAPERS;20
9.1;CHAPTER 1. THE METHOD OF ALTERNATING CHAINS;20
9.1.1;0. Introduction;20
9.1.2;1. An axiomatic approach to hierarchies;22
9.1.3;2. Difference hierarchies;24
9.1.4;3. Alternating chains;31
9.1.5;4. Applications;34
9.2;CHAPTER 2. SEMANTIC CONSTRUCTION OF LEWIS'S SYSTEMS S4 AND S5;36
9.2.1;1. Introduction;36
9.2.2;2. Formulas;36
9.2.3;3. Semantics;36
9.2.4;4. Sequents;37
9.2.5;5. Closure and reduction schemata; semantic tableaux;37
9.2.6;6. Axiomatization;39
9.2.7;7. S4 in terms of strict implication;40
9.2.8;8. Further metamathematical results;41
9.2.9;9. S4 and topology;42
9.2.10;10. The System S5;43
9.3;CHAPTER 3. CONTINUOUS MODEL THEORY;44
9.3.1;Introduction;44
9.3.2;1. 2-valued model theory;45
9.3.3;2. X-valued model theory;47
9.3.4;3. Continuous model theory;49
9.3.5;4. Adequate model theory;52
9.3.6;5. Ordered model theory;54
9.4;CHAPTER 4. INDEPENDENCE RESULTS IN SET THEORY;58
9.5;CHAPTER 5. BOOLEAN NOTIONS EXTENDED TO HIGHER DIMENSIONS;74
9.5.1;1. Introduction and Summary;74
9.5.2;2. Operations;77
9.5.3;3. Fields;85
9.5.4;4.
Algebras;86
9.5.5;5. Semilocal and elementary operations;87
9.6;CHAPTER 6. ELEMENTARY THEORIES WITH MODELS WITHOUT
AUTOMORPHISMS;89
9.7;CHAPTER 7. COMBINATORIAL THEOREMS FOR THE CONSTRUCTION OF MODELS;96
9.7.1;1. Categorical truth-definitions;97
9.7.2;2. Compactness;101
9.7.3;3. Using compactness;105
9.8;CHAPTER 8. SOME APPLICATIONS OF THE NOTIONS OF FORCING AND GENERIC SETS (SUMMARY);108
9.9;CHAPTER 9. A HYPOTHESIS CONCERNING THE EXTENSION OF FINITE RELATIONS AND ITS VERIFICATION FOR CERTAIN SPECIAL CASES;115
9.9.1;1. Multirelation; restriction, extension, isomorphism;115
9.9.2;2.
p-equivalence;117
9.9.3;3. The hypothesis; its verification for unary multirelations and for binary multirelations with p = 2;119
9.9.4;4. The hypothesis for binary multirelations with arbitrary p can be reduced to the case p = 3;121
9.10;CHAPTER 10. THE THEORIES OF FUNCTORS AND MODELS;126
9.10.1;Part Two. Applications to Model Theory;132
9.11;CHAPTER 11. LANGUAGES WITH ADDED QUANTIFIER "THERE EXIST AT LEAST
.a";140
9.12;CHAPTER 12. MODEL-THEORETIC METHODS IN THE STUDY OF ELEMENTARY LOGIC;151
9.12.1;1. Isomorphism of theories;153
9.12.2;2. The main theorem;154
9.12.3;3. Conclusions and problems;163
9.13;CHAPTER 13. EXTENSIONS OF RELATIONAL STRUCTURES;165
9.13.1;1. Introduction;165
9.13.2;2. The embedding property and the amalgamation property;166
9.13.3;3. Elementary consequences of the amalgamation property;168
9.13.4;4. Free products of algebras;169
9.13.5;5. Homogeneous-universal structures;171
9.13.6;6. Algebraic extensions of relational structures;172
9.14;CHAPTER 14. FINITE APPROXIMATIONS OF INFINITELY LONG FORMULAS;177
9.14.1;1. Preliminaries;178
9.14.2;2. Conjunctive formulas;182
9.14.3;3. Admissible formulas;185
9.15;CHAPTER 15. TOPICS IN THE THEORY OF DEFINITION;189
9.16;CHAPTER 16. LOGICAL STRUCTURES ARISING IN QUANTUM THEORY;196
9.17;CHAPTER 17. MODEL-THEORETIC INVARIANTS: APPLICATIONS
TO RECURSIVE AND HYPERARITHMETIC OPERATIONS;209
9.17.1;Introduction;209
9.17.2;1. Basic Notions;211
9.17.3;2. Invariants;215
9.18;CHAPTER 18. SEMANTICAL ANALYSIS OF MODAL LOGIC II.
NON-NORMAL MODAL PROPOSITIONAL CALCULI;225
9.18.1;1. Generalities; Halldén's property;225
9.18.2;2. Propositional calculi considered;227
9.18.3;3. Models.;229
9.18.4;4. Semantic tableaux;231
9.18.5;5.1. Consistency property;233
9.18.6;6. Applications.;236
9.18.7;7. Other
systems;239
9.19;CHAPTER 19. THE FRAENKEL-MOSTOWSKI METHOD FOR INDEPENDENCE
PROOFS IN SET THEORY;240
9.19.1;1. Introduction;240
9.19.2;2. One body of results;241
9.19.3;3. Survey of additional results and problems;243
9.19.4;4. A few detailed results;244
9.20;CHAPTER 20. FREE PRODUCT IN GENERAL ALGEBRAS;248
9.21;CHAPTER 21. MODEL-THEORETIC METHODS AND RESULTS IN THE THEORY OF CYLINDRIC
ALGEBRAS;257
9.21.1;1. Representable cylindric algebras;257
9.21.2;2. Non-finite
axiomatizability;262
9.22;CHAPTER 22. REDUCTIONS OF HIGHER-ORDER LOGIC;270
9.22.1;1. Preliminaries;270
9.22.2;2. Definition of the
reduction;277
9.22.3;3. Spectrum problems;278
9.22.4;4. Logical truth;280
9.23;CHAPTER 23. OMITTING CLASSES OF ELEMENTS;284
9.24;CHAPTER 24. UNIVERSAL GROUPS OF AUTOMORPHISMS OF MODELS;293
9.24.1;Introduction;293
9.24.2;1. The class G(.)
and its properties;294
9.24.3;2. Algebraic characterization of universal groups;297
9.24.4;3. Elementary characterization of universal groups;299
9.25;CHAPTER 25. TOPICS IN NON-ARCHIMEDEAN MATHEMATICS;304
9.25.1;1· Introduction;304
9.25.2;2. Non-standard analysis of real numbers;305
9.25.3;3. Higher-order non-standard analysis;306
9.25.4;4. Application to topological spaces;311
9.25.5;5. Application to normed linear spaces;314
9.25.6;6. Methodological considerations;316
9.26;CHAPTER 26. THE DECISION PROBLEM FOR FIELDS;318
9.26.1;1. Introduction;318
9.26.2;2. Decidable fields;319
9.26.3;3. Undecidable fields;322
9.27;CHAPTER 27. ON MODELS OF ELEMENTARY ELLIPTIC GEOMETRY;331
9.27.1;Introduction;331
9.27.2;1. Axioms;332
9.27.3;2. Point calculus and coordinates;333
9.27.4;3. Elliptic trigonometry;336
9.27.5;4. Representation theorems;345
9.28;CHAPTER 28. LOGIC WITH DENUMERABLY LONG FORMULAS
AND FINITE STRINGS OF QUANTIFIERS;348
9.29;CHAPTER 29. NON-STANDARD MODELS FOR FRAGMENTS OF NUMBER THEORY;361
9.29.1;Summary;361
9.29.2;1. Introduction;361
9.29.3;2. Rules and axioms of induction;363
9.29.4;3. Induction-free equivalents for systems up to and including multiplication;368
9.29.5;4. A non-standard model for the system
including;372
9.29.6;5. Inclusion of the functions
[x/n];375
9.29.7;6. Other forms of free-variable induction;376
9.30;CHAPTER 30. APPLICATIONS OF MODEL THEORY TO DEGREES OF UNSOLVABILITY;378
9.31;CHAPTER 31. LOGICS APPROPRIATE TO EMPIRICAL THEORIES;383
9.31.1;1. Introduction;383
9.31.2;2. Principles of invariance and three-valued logic;383
9.31.3;3. Families of Boolean algebras in quantum mechanics;388
9.32;CHAPTER 32. ON THE DENUMERABLE MODELS OF THEORIES WITH EXTRA PREDICATES;395
9.32.1;1. General terminology;395
9.32.2;2. Theories with extra
predicates;396
9.32.3;3. Theories with extra predicates of special form;397
9.32.4;4. Lemmas for Theorem 2;399
9.32.5;5. Proof of Theorem 2;401
9.32.6;6. Uniqueness of
T;404
9.32.7;7. Application to Craig's lemma;407
9.33;CHAPTER 33. A LÖWENHEIM-SKOLEM THEOREM FOR CARDINALS FAR APART;409
9.33.1;1. Introduction;409
9.33.2;2. Preliminaries;410
9.33.3;3. Summary of known results;411
9.33.4;4. The main theorem and its consequences;413
9.33.5;5. Proof of the main theorem;416
10;PART II: CONTRIBUTED PAPERS;421
10.1;CHAPTER 34. SYNONYMOUS THEORIES;421
10.2;CHAPTER 35. FINITE-QUANTIFIER EQUIVALENCE;426
10.2.1;1. A characterization of equivalence with respect to finite-quantifier
sentences of rank d;427
10.2.2;2. An application to well-orderings;430
10.3;CHAPTER 36. ALGEBRAIC THEORIES, ALGEBRAIC CATEGORIES, AND ALGEBRAIC FUNCTORS;432
10.4;CHAPTER 37. EXTENSIVE ULTRAPRODUCTS AND HAAR MEASURE;438
10.4.1;1. Extensive ultraproducts and summation;438
10.4.2;2. The measure;440
10.4.3;3. Integration;442
11;PART III: ABSTRACTS;443
11.1;CHAPTER 38. RELATIVELY CATEGORICAL AND NORMAL THEORIES;443
11.2;CHAPTER 39. FREE STRUCTURES AND CATEGORIES;446
11.3;CHAPTER 40. FINITE-DIMENSIONAL ANALOGUES TO BOOLEAN ALGEBRAS;448
11.4;CHAPTER 41. BOOLEAN RECURSIVE FUNCTIONS AND CLOSURE ALGEBRA;450
11.5;CHAPTER 42. A UNIFYING PRINCIPLE IN QUANTIFICATION THEORY;452
11.6;CHAPTER 43. 2
CAN BE ANYTHING IT OUGHT TO BE;454
11.7;CHAPTER 44. CONSTRUCTION OF A MODEL FOR GÖDEL-BERNAYS SET THEORY FOR WHICH THE CLASS OF NATURAL NUMBERS IS A SET OF THE MODEL AND A PROPER CLASS IN THE THEORY;455
12;PART IV: BIBLIOGRAPHY WITH EXPLANATORY NOTES;457
12.1;SOME NOTES ON THE THEORY OF MODELS;457
12.2;A BIBLIOGRAPHY OF THE THEORY OF MODELS;461
12.3;LIST OF REGISTERED PARTICIPANTS;512