Roads to Infinity: The Mathematics of Truth and Proof

Roads to Infinity: The Mathematics of Truth and Proof

by John C. Stillwell
     
 

ISBN-10: 1568814666

ISBN-13: 9781568814667

Pub. Date: 07/13/2010

Publisher: Taylor & Francis

Winner of a CHOICE Outstanding Academic Title Award for 2011!

This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and

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Overview

Winner of a CHOICE Outstanding Academic Title Award for 2011!

This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and partly informal, but with due attention to the subtleties of the subject.

Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Gödel.

Product Details

ISBN-13:
9781568814667
Publisher:
Taylor & Francis
Publication date:
07/13/2010
Pages:
250
Product dimensions:
6.00(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface

1 The Diagonal Argument 1

1.1 Counting and Countability 2

1.2 Does One Infinite Size Fit All? 4

1.3 Cantor's Diagonal Argument 6

1.4 Transcendental Numbers 10

1.5 Other Uncountability Proofs 12

1.6 Rates of Growth 14

1.7 The Cardinality of the Continuum 16

1.8 Historical Background 19

2 Ordinals 29

2.1 Counting Past Infinity 30

2.2 The Countable Ordinals 33

2.3 The Axiom of Choice 37

2.4 The Continuum Hypothesis 40

2.5 Induction 42

2.6 Cantor Normal Form 46

2.7 Goodstein's Theorem 47

2.8 Hercules and the Hydra 51

2.9 Historical Background 54

3 Computability and Proof 67

3.1 Formal Systems 68

3.2 Post's Approach to Incompleteness 72

3.3 Godel's First Incompleteness Theorem 75

3.4 Godel's Second Incompleteness Theorem 80

3.5 Formalization of Computability 82

3.6 The Halting Problem 85

3.7 The Entscheidungsproblem 87

3.8 Historical Background 89

4 Logic 97

4.1 Propositional Logic 98

4.2 A Classical System 100

4.3 A Cut-Free System for Propositional Logic 102

4.4 Happy Endings 105

4.5 Predicate Logic 106

4.6 Completeness, Consistency, Happy Endings 110

4.7 Historical Background 112

5 Arithmetic 119

5.1 How Might We Prove Consistency? 120

5.2 Formal Arithmetic 121

5.3 The Systems PA and PAw 122

5.4 Embedding PA in PAw 124

5.5 Cut Elimination in PAw 127

5.6 The Height of This Great Argument 130

5.7 Roads to Infinity 133

5.8 Historical Background 135

6 Natural Unprovable Sentences 139

6.1 A Generalized Goodstein Theorem 140

6.2 Countable Ordinals via Natural Numbers 141

6.3 From Generalized Goodstein to Well-Ordering 144

6.4 Generalized and Ordinary Goodstein 146

6.5 Provably Computable Functions 147

6.6 Complete Disorder is Impossible 151

6.7 The Hardest Theorem in Graph Theory 154

6.8 Historical Background 157

7 Axioms of Infinity 165

7.1 Set Theory without Infinity 165

7.2 Inaccessible Cardinals 168

7.3 The Axiom of Determinacy 170

7.4 Largeness Axioms for Arithmetic 172

7.5 Large Cardinals and Finite Mathematics 173

7.6 Historical Background 177

Bibliography 183

Index 189

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