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Cambridge University Press
Subsystems of Second Order Arithmetic / Edition 2

Subsystems of Second Order Arithmetic / Edition 2

by Stephen G. Simpson


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Subsystems of Second Order Arithmetic / Edition 2

Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics.

In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second order arithmetic. Additional results are presented in an appendix.

Product Details

ISBN-13: 9780521150149
Publisher: Cambridge University Press
Publication date: 01/28/2010
Series: Perspectives in Logic Series
Edition description: New Edition
Pages: 464
Product dimensions: 6.10(w) x 9.10(h) x 1.10(d)

Table of Contents

List of Tables xi

Preface xiii

Acknowledgments xv

Chapter I Introduction 1

I.1 The Main Question 1

I.2 Subsystems of Z2 2

I.3 The System ACA0 6

I.4 Mathematics within ACA0 9

I.5 $$-CA0 and Stronger Systems 16

I.6 Mathematics within $$-CA0 19

I.7 The System RCA0 23

I.8 Mathematics within RCA0 27

I.9 Reverse Mathematics 32

I.10 The System WKL0 35

I.11 The System ATR0 38

I.12 The Main Question, Revisited 42

I.13 Outline of Chapters II through X 43

I.14 Conclusions 60

Part A Development of Mathematics within Subsystems of Z2

Chapter II Recursive Comprehension 63

II.1 The Formal System RCA0 63

II.2 Finite Sequences 65

II.3 Primitive Recursion 69

II.4 The Number Systems 73

II.5 Complete Separable Metric Spaces 78

II.6 Continuous Functions 84

II.7 More on Complete Separable Metric Spaces 88

II.8 Mathematical Logic 92

II.9 Countable Fields 96

II.10 Separable Banach Spaces 99

II.11 Conclusions 103

Chapter III Arithmetical Comprehension 105

III.1 The Formal System ACA0 105

III.2 Sequential Compactness 106

III.3 Strong Algebraic Closure 110

III.4 Countable Vector Spaces 112

III.5 Maximal Ideals in Countable Commutative Rings 115

III.6 Countable Abelian Groups 118

III.7 Köet;nig's Lemma and Ramsey's Theorem 121

III.8 Conclusions 125

Chapter IV Weak Köet;nig's Lemma 127

IV.1 The Heine/Borel Covering Lemma 127

IV.2 Properties of Continuous Functions 133

IV.3 The Göet;del Completeness Theorem 139

IV.4 Formally Real Fields 141

IV.5 Uniqueness of Algebraic Closure 144

IV.6 Prime Ideals in Countable Commutative Rings 146

IV.7 Fixed Point Theorems 149

IV.8 Ordinary Differential Equations 154

IV.9The Separable Hahn/Banach Theorem 160

IV.10 Conclusions 165

Chapter V Arithmetical Transfinite Recursion 167

V.1 Countable Well Orderings; Analytic Sets 167

V.2 The Formal System ATR0 173

V.3 Borel Sets 178

V.4 Perfect Sets; Pseudohierarchies 185

V.5 Reversals 189

V.6 Comparability of Countable Well Orderings 195

V.7 Countable Abelian Groups 199

V.8 σ01 and δ01 Determinacy 203

V.9 The σ01 and δ01 Ramsey Theorems 210

V.10 Conclusions 215

Chapter VI $$ Comprehension 217

VI.1 Perfect Kernels 217

VI.2 Coanalytic Uniformization 221

VI.3 Coanalytic Equivalence Relations 225

VI.4 Countable Abelian Groups 230

VI.5 σ01 λ $$ Determinacy 232

VI.6 The δ02 Ramsey Theorem 236

VI.7 Stronger Set Existence Axioms 239

VI.8 Conclusions 240

Part B Models of Subsystems of Z2

Chapter VII β-Models 243

VII.1 The Minimum β-Model of $$-CA0 244

VII.2 Countable Coded β-Models 248

VII.3 A Set-Theoretic Interpretation of ATR0 258

VII.4 Constructible Sets and Absoluteness 272

VII.5 Strong Comprehension Schemes 286

VII.6 Strong Choice Schemes 294

VII.7 β-Model Reflection 303

VII.8 Conclusions 307

Chapter VIII ω-Models 309

VIII.1 ω-Models of RCA0 and ACA0 310

VIII.2 Countable Coded ω-Models of WKL0 314

VIII.3 Hyperarithmetical Sets 322

VIII.4 ω-Models of σ11 Choice 333

VIII.5 ω-Model Reflection and Incompleteness 342

VIII.6 ω-Models of Strong Systems 348

VIII.7 Conclusions 356

Chapter IX Non-ω-Models 359

IX.1 The First Order Parts of RCA0 and ACA0 360

IX.2 The First Order Part of WKL0 365

IX.3 A Conservation Result for Hilbert's Program 369

IX.4 Saturated Models 379

IX.5 Gentzen-Style Proof Theory 386

IX.6 Conclusions 388


Chapter X Additional Results 391

X.1 Measure Theory 391

X.2 Separable Banach Spaces 396

X.3 Countable Combinatorics 399

X.4 Reverse Mathematics for RCA0 405

X.5 Conclusions 407

Bibliography 409

Index 425

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