Through a series of case studies, this volume examines these axioms to prove particular theorems in core mathematical areas.
COVER; HALF-TITLE; SERIES-TITLE; TITLE; COPYRIGHT; CONTENTS; LIST OF TABLES; PREFACE; ACKNOWLEDGMENTS; Chapter I INTRODUCTION; I.1. The Main Question; I.2. Subsystems of Z2; I.3. The System ACA0; I.4. Mathematics within ACA0; I.5. Pi11 -CA0 and Stronger Systems; I.6. Mathematics within Pi11 -CA0; I.7. The System RCA0; I.8. Mathematics within RCA0; I.9. Reverse Mathematics; I.10. The System WKL0; I.11. The System ATR0; I.12. The Main Question, Revisited; I.13. Outline of Chapters II through X; I.14. Conclusions; Part A DEVELOPMENT OF MATHEMATICS WITHIN SUBSYSTEMS OF Z2. Chapter II RECURSIVE COMPREHENSIONChapter III ARITHMETICAL COMPREHENSION; Chapter IV WEAK KNIG'S LEMMA; Chapter V ARITHMETICAL TRANSFINITE RECURSION; Chapter VI Pi11 COMPREHENSION; Part B MODELS OF SUBSYSTEMS OF Z2; Chapter VII beta-MODELS; Chapter VIII omega-MODELS; Chapter IX NON-omega-MODELS; APPENDIX; Chapter X ADDITIONAL RESULTS; BIBLIOGRAPHY; INDEX.
Author(s): Stephen G. Simpson
Series: Perspectives in Logic
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 462
Contents......Page 9
List of Tables......Page 13
Preface......Page 15
1. The Main Question......Page 19
2. Subsystems of Z_2......Page 20
3. The System ACA_0......Page 24
4. Mathematics within ACA_0......Page 27
5. Π^1_1 - CA_0 and Stronger Systems......Page 34
6. Mathematics within Π^1_1 - CA_0......Page 37
7. The System RCA_0......Page 41
8. Mathematics within RCA_0......Page 45
9. Reverse Mathematics......Page 50
10. The System WKL_0......Page 53
11. The System ATR_0......Page 56
12. The Main Question, Revisited......Page 60
13. Outline of Chapters II through X......Page 61
14. Conclusions......Page 78
Part A. Development of Mathematics within Subsystems of Z_2......Page 79
1. The Formal System RCA_0......Page 81
2. Finite Sequences......Page 83
3. Primitive Recursion......Page 87
4. The Number Systems......Page 91
5. Complete Separable Metric Spaces......Page 96
6. Continuous Functions......Page 102
7. More on Complete Separable Metric Spaces......Page 106
8. Mathematical Logic......Page 110
9. Countable Fields......Page 114
10. Separable Banach Spaces......Page 117
11. Conclusions......Page 121
1. The Formal System ACA_0......Page 123
2. Sequential Compactness......Page 124
3. Strong Algebraic Closure......Page 128
4. Countable Vector Spaces......Page 130
5. Maximal Ideals in Countable Commutative Rings......Page 133
6. Countable Abelian Groups......Page 136
7. König’s Lemma and Ramsey’s Theorem......Page 139
8. Conclusions......Page 143
1. The Heine/Borel Covering Lemma......Page 145
2. Properties of Continuous Functions......Page 151
3. The Gödel Completeness Theorem......Page 157
4. Formally Real Fields......Page 159
5. Uniqueness of Algebraic Closure......Page 162
6. Prime Ideals in Countable Commutative Rings......Page 164
7. Fixed Point Theorems......Page 167
8. Ordinary Di»erential Equations......Page 172
9. The Separable Hahn/Banach Theorem......Page 178
10. Conclusions......Page 183
1. Countable Well Orderings; Analytic Sets......Page 185
2. The Formal System ATR_0......Page 191
3. Borel Sets......Page 196
4. Perfect Sets; Pseudohierarchies......Page 203
5. Reversals......Page 207
6. Comparability of Countable Well Orderings......Page 213
7. Countable Abelian Groups......Page 217
8. Σ^0_1 and Delta01 Determinacy......Page 221
9. The Σ^0_1 and Δ^0_1 Ramsey Theorems......Page 228
10. Conclusions......Page 233
1. Perfect Kernels......Page 235
2. Coanalytic Uniformization......Page 239
3. Coanalytic Equivalence Relations......Page 243
4. Countable Abelian Groups......Page 248
5. …Determinacy......Page 250
6. The Δ^0_2 Ramsey Theorem......Page 254
7. Stronger Set Existence Axioms......Page 257
8. Conclusions......Page 258
Part B. Models of Subsystems of Z_2......Page 259
VII. β-Models......Page 261
1. The Minimum β-Model of Π^1_1 - CA_0......Page 262
2. Countable Coded β-Models......Page 266
3. A Set-Theoretic Interpretation of ATR_0......Page 276
4. Constructible Sets and Absoluteness......Page 290
5. Strong Comprehension Schemes......Page 304
6. Strong Choice Schemes......Page 312
7. β-Model Reflection......Page 321
8. Conclusions......Page 325
VIII. ω-Models......Page 327
1. ω-Models of RCA_0 and ACA_0......Page 328
2. Countable Coded ω-Models of WKL_0......Page 332
3. Hyperarithmetical Sets......Page 340
4. ω-Models of Σ_1 Choice......Page 351
5. ω-Model Reflection and Incompleteness......Page 360
6. ω-Models of Strong Systems......Page 366
7. Conclusions......Page 374
IX. Non-ω-Models......Page 377
1. The First Order Parts of RCA_0 and ACA_0......Page 378
2. The First Order Part of WKL_0......Page 383
3. A Conservation Result for Hilbert’s Program......Page 387
4. SaturatedModels......Page 397
5. Gentzen-Style Proof Theory......Page 404
6. Conclusions......Page 406
Appendix......Page 407
1. Measure Theory......Page 409
2. Separable Banach Spaces......Page 414
3. Countable Combinatorics......Page 417
4. Reverse Mathematics for RCA_0......Page 423
5. Conclusions......Page 425
Bibliography......Page 427
Index......Page 443
