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| List of Tables | |
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| Preface | |
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| Acknowledgments | |
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| Introduction | |
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| The Main Question | |
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| Subsystems of Z2 | |
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| The System ACA0 | |
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| Mathematics within ACA0 | |
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| $$-CA0 and Stronger Systems | |
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| Mathematics within $$-CA0 | |
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| The System RCA0 | |
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| Mathematics within RCA0 | |
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| Reverse Mathematics | |
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| The System WKL0 | |
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| The System ATR0 | |
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| The Main Question, Revisited | |
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| Outline of Chapters II through X | |
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| Conclusions | |
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| Development of Mathematics within Subsystems of Z2 | |
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| Recursive Comprehension | |
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| The Formal System RCA0 | |
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| Finite Sequences | |
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| Primitive Recursion | |
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| The Number Systems | |
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| Complete Separable Metric Spaces | |
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| Continuous Functions | |
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| More on Complete Separable Metric Spaces | |
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| Mathematical Logic | |
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| Countable Fields | |
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| Separable Banach Spaces | |
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| Conclusions | |
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| Arithmetical Comprehension | |
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| The Formal System ACA0 | |
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| Sequential Compactness | |
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| Strong Algebraic Closure | |
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| Countable Vector Spaces | |
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| Maximal Ideals in Countable Commutative Rings | |
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| Countable Abelian Groups | |
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| Köet;nig's Lemma and Ramsey's Theorem | |
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| Conclusions | |
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| Weak Köet;nig's Lemma | |
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| The Heine/Borel Covering Lemma | |
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| Properties of Continuous Functions | |
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| The Göet;del Completeness Theorem | |
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| Formally Real Fields | |
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| Uniqueness of Algebraic Closure | |
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| Prime Ideals in Countable Commutative Rings | |
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| Fixed Point Theorems | |
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| Ordinary Differential Equations | |
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| The Separable Hahn/Banach Theorem | |
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| Conclusions | |
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| Arithmetical Transfinite Recursion | |
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| Countable Well Orderings; Analytic Sets | |
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| The Formal System ATR0 | |
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| Borel Sets | |
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| Perfect Sets; Pseudohierarchies | |
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| Reversals | |
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| Comparability of Countable Well Orderings | |
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| Countable Abelian Groups | |
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| �01 and �01 Determinacy | |
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| The �01 and �01 Ramsey Theorems | |
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| Conclusions | |
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| $$ Comprehension | |
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| Perfect Kernels | |
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| Coanalytic Uniformization | |
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| Coanalytic Equivalence Relations | |
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| Countable Abelian Groups | |
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| �01 � $$ Determinacy | |
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| The �02 Ramsey Theorem | |
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| Stronger Set Existence Axioms | |
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| Conclusions | |
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| Models of Subsystems of Z2 | |
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| �-Models | |
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| The Minimum �-Model of $$-CA0 | |
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| Countable Coded �-Models | |
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| A Set-Theoretic Interpretation of ATR0 | |
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| Constructible Sets and Absoluteness | |
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| Strong Comprehension Schemes | |
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| Strong Choice Schemes | |
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| �-Model Reflection | |
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| Conclusions | |
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| �-Models | |
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| �-Models of RCA0 and ACA0 | |
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| Countable Coded �-Models of WKL0 | |
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| Hyperarithmetical Sets | |
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| �-Models of �11 Choice | |
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| �-Model Reflection and Incompleteness | |
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| �-Models of Strong Systems | |
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| Conclusions | |
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| Non-�-Models | |
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| The First Order Parts of RCA0 and ACA0 | |
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| The First Order Part of WKL0 | |
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| A Conservation Result for Hilbert's Program | |
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| Saturated Models | |
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| Gentzen-Style Proof Theory | |
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| Conclusions | |
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| Additional Results | |
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| Measure Theory | |
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| Separable Banach Spaces | |
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| Countable Combinatorics | |
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| Reverse Mathematics for RCA0 | |
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| Conclusions | |
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| Bibliography | |
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| Index | |