Mathematical Logic in the 20th Century

Gerald E. Sacks9789810247362, 981-02-4736-2

This anthology contains 31 significant papers in mathematical logic, mostly published in the second half of the twentieth century. Sacks, selecting on the basis of brevity as well as results, includes the work of Godel (“Consistency-Proof for the Generalized Continuum- Hypothesis”), Kleene (“Recursive Functionals and Quantifiers of Finite Types”), Tarski (“A Decision Method for Elementary Algebra and Geometry”), Robinson (“Non-Standard Analysis”), Kriesel (“Model- Theoretic Invariants”), Cohen (both parts of “The Independence of the Continuum Hypothesis”), Morley (“Categoricity in Power”), Shelah (“Stable Theories”), Hrushovski (“The Mordell-Lang Conjecture for Function Fields”) and Woodin (“Supercompact Cardinals, Sets of Reals, and Weakly Homogeneous Trees”).

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Table of contents :
Contents……Page 12
THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS……Page 16
THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS, II*……Page 22
§ 0 Introduction……Page 28
§ 1 The approach……Page 30
§ 2 Fine structure lemmas……Page 32
§ 3 The non w cofinal case……Page 41
§ 4 Vicious sequences……Page 47
§ 5 The w cofinal case……Page 52
Bibliography……Page 55
THREE THEOREMS ON RECURSIVE ENUMERATION. I. DECOMPOSITION. II. MAXIMAL SET. III. ENUMERATION WITHOUT DUPLICATION……Page 56
BIBLIOGRAPHY……Page 63
Introduction……Page 64
Section 1……Page 68
Section 2……Page 79
Section 3……Page 88
Section 4……Page 95
References……Page 96
INTRODUCTION TO -LOGIC……Page 97
1. SYSTEMS OF DENOTATIONS……Page 98
2. DILATORS……Page 101
3. THE ALGEBRAIC THEORY OF DILATORS……Page 103
4. DILATORS AS WELL-ORDERED CLASSES……Page 106
5. A AND TRADITIONAL PROOF-THEORY……Page 110
6. ß-PROOFS……Page 113
7. INDUCTIVE LOGIC……Page 115
8. APPLICATIONS TO GENERALIZED RECURSION……Page 117
9. DESCRIPTIVE SET-THEORY……Page 119
2. Relations to Generalized Recursion……Page 120
5. A and Related Topics……Page 121
CONSISTENCY-PROOF FOR THE GENERALIZED CONTINUUM-HYPOTHESIS¹……Page 123
1. INTRODUCTION……Page 128
2. THIN TYPES AND ZARISKI GEOMETRIES……Page 131
3. MANIN’S HOMOMORPHISM AND BUIUM’S REDUCTION……Page 139
4. ABELIAN GROUPS OF FINITE MORLEY DIMENSION……Page 140
5. THE MAIN THEOREM……Page 145
6. A QUESTION OF VOLOCH’S……Page 147
REFERENCES……Page 150
Introduction…….Page 152
1. Basic Notions…….Page 154
2. Invariants…….Page 158
2.1. Technical lemmas on general models…….Page 160
2.2. Technical lemmas on w-models…….Page 161
3.1. Theory of w-models…….Page 163
3.2. Recursion theory on the recursive ordinals…….Page 164
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF FINITE TYPES I……Page 168
1. Primitive recursive functions…….Page 169
2. Alterations of quantifiers…….Page 174
3. Partial and general recursive functions…….Page 177
4. Construction of indices…….Page 184
5. Reduction of the inductive definition of {z} (a)~w to an explicit definition…….Page 189
6. Reduction in type of a quantifier…….Page 202
7. Predicates of order r…….Page 206
8. µ-recursiveness versus general recursiveness…….Page 211
BIBLIOGRAPHY……Page 218
0. Introduction……Page 220
1. Formulation of the problem……Page 221
2. The priority tree and the construction……Page 223
3. Properties of the construction……Page 234
4. Verification of Propositions 1 to 10……Page 246
References……Page 278
ENUMERABLE SETS ARE DIOPHANTINE……Page 284
ADDENDUM *……Page 287
BIBLIOGRAPHY……Page 288
Introduction…….Page 289
1. Preliminaries…….Page 290
2. Transcendence in rank…….Page 293
3. Results depending on Ramsey’s theorem…….Page 300
4. Models of totally transcendental theories…….Page 304
5. Saturated models and categoricity in power…….Page 309
REFERENCES……Page 313
HYPERANALYTIC PREDICATES……Page 314
1. A hyperanalytic function not recursive in any H 2 (a)…….Page 315
2. Functions recursive in a type-3 object…….Page 319
3. The hyperanalytic hierarchy…….Page 324
4. Minimum functions…….Page 329
5. Predicates r.e. in ³E…….Page 334
6. Normal forms for predicates r.e. in ³E…….Page 338
7. An extension of the hyperanalytic hierarchy…….Page 344
8. Comments on results for types other than 3…….Page 346
BIBLIOGRAPHY……Page 347
Introduction……Page 348
§ 1. Sequences and Quasisequences……Page 351
§ 2. Functional Representation of Operators……Page 352
§ 3. A Universal Partial Recursive Operator……Page 354
§ 4. The Calculus of M-problems……Page 355
§ 2. Post’s Reduction Problem……Page 357
BIBLIOGRAPHY……Page 365
Introduction…….Page 367
1. Recursive versus recursively enumerable sets…….Page 373
2. A form of Godel’s theorem…….Page 375
3. The complete set K; creative sets…….Page 378
4. One-one reductibility, to K; many-one reducibility…….Page 379
5. Simple sets…….Page 381
6. Reducibility by truth-tables…….Page 382
7. Non-reducibility of creative sets to simple sets by bounded truth-tables…….Page 384
8. Counter-example for unbounded truth-tables…….Page 387
9. Hyper-simple sets…….Page 388
10. Non-reducibility of creative sets to hyper-simple sets by truth-tables unrestricted…….Page 391
11. General (Turing) reducibility…….Page 394
BIBLIOGRAPHY……Page 398
1. Introduction…….Page 400
2. Non-standard analysis and non-archimedean fields…….Page 401
3. Examples in non-standard Analysis…….Page 403
REFERENCES……Page 408
0. Introduction……Page 426
1. Notations…….Page 427
2. On possible cardinalities of S(A)……Page 429
3. On some properties of stable theories…….Page 433
4. On categorical elementary and pseudo elementary classes…….Page 437
REFERENCES……Page 440
THE PROBLEM OF PREDICATIVITY……Page 442
REFERENCES……Page 449
1. Relations involving both join and jump operations……Page 551
2. Sets of degrees without g.l.b. or without l.u.b…….Page 554
3. Non-density……Page 556
REFERENCES……Page 562
INTRODUCTION……Page 563
SECTION 1. THE SYSTEM OF ELEMENTARY ALGEBRA……Page 568
SECTION 2. DECISION METHOD FOR ELEMENTARY ALGEBRA……Page 577
SECTION 3. EXTENSIONS TO RELATED SYSTEMS……Page 605
NOTES……Page 608
BIBLIOGRAPHY……Page 619
SUPPLEMENTARY NOTES……Page 621
1. INTRODUCTION……Page 643
First Main Theorem…….Page 645
Second Main Theorem…….Page 646
2. TOWARDS THE PROOF OF THE FIRST MAIN THEOREM……Page 647
3. RESULTS OF KHOVANSKII AND VAN DEN DRIES……Page 652
4. DlFFERENTIABLE GERMS IN ARBITRARY EXPANSIONS OF R……Page 654
5. DEFINABLE POINTS ON COMPONENTS AND THE PROOF OF LEMMA 2.7……Page 659
6. ONE DIMENSIONAL VARIETIES……Page 662
7. THE PROOF OF LEMMA 2.8……Page 665
8. THE PROOF OF LEMMA 2.9……Page 670
9. TOWARDS THE PROOF OF THE SECOND MAIN THEOREM……Page 675
10. SMOOTH 0-MINIMAL THEORIES……Page 677
11. BOUNDING THE SOLUTIONS TO EXPONENTIAL-POLYNOMIAL EQUATIONS AND THE COMPLETION OF THE PROOF OF THE SECOND MAIN THEOREM……Page 683
REFERENCES……Page 685
1…….Page 692
2. Proofs……Page 694
References……Page 705
Permissions……Page 706