| List of Tables | p. xi |
| Preface | p. xiii |
| Acknowledgments | p. xv |
| Introduction | p. 1 |
| The Main Question | p. 1 |
| Subsystems of Z2 | p. 2 |
| The System ACA0 | p. 6 |
| Mathematics within ACA0 | p. 9 |
| $$-CA0 and Stronger Systems | p. 16 |
| Mathematics within $$-CA0 | p. 19 |
| The System RCA0 | p. 23 |
| Mathematics within RCA0 | p. 27 |
| Reverse Mathematics | p. 32 |
| The System WKL0 | p. 35 |
| The System ATR0 | p. 38 |
| The Main Question, Revisited | p. 42 |
| Outline of Chapters II through X | p. 43 |
| Conclusions | p. 60 |
| Development of Mathematics within Subsystems of Z2 | |
| Recursive Comprehension | p. 63 |
| The Formal System RCA0 | p. 63 |
| Finite Sequences | p. 65 |
| Primitive Recursion | p. 69 |
| The Number Systems | p. 73 |
| Complete Separable Metric Spaces | p. 78 |
| Continuous Functions | p. 84 |
| More on Complete Separable Metric Spaces | p. 88 |
| Mathematical Logic | p. 92 |
| Countable Fields | p. 96 |
| Separable Banach Spaces | p. 99 |
| Conclusions | p. 103 |
| Arithmetical Comprehension | p. 105 |
| The Formal System ACA0 | p. 105 |
| Sequential Compactness | p. 106 |
| Strong Algebraic Closure | p. 110 |
| Countable Vector Spaces | p. 112 |
| Maximal Ideals in Countable Commutative Rings | p. 115 |
| Countable Abelian Groups | p. 118 |
| Köet;nig's Lemma and Ramsey's Theorem | p. 121 |
| Conclusions | p. 125 |
| Weak Köet;nig's Lemma | p. 127 |
| The Heine/Borel Covering Lemma | p. 127 |
| Properties of Continuous Functions | p. 133 |
| The Göet;del Completeness Theorem | p. 139 |
| Formally Real Fields | p. 141 |
| Uniqueness of Algebraic Closure | p. 144 |
| Prime Ideals in Countable Commutative Rings | p. 146 |
| Fixed Point Theorems | p. 149 |
| Ordinary Differential Equations | p. 154 |
| The Separable Hahn/Banach Theorem | p. 160 |
| Conclusions | p. 165 |
| Arithmetical Transfinite Recursion | p. 167 |
| Countable Well Orderings; Analytic Sets | p. 167 |
| The Formal System ATR0 | p. 173 |
| Borel Sets | p. 178 |
| Perfect Sets; Pseudohierarchies | p. 185 |
| Reversals | p. 189 |
| Comparability of Countable Well Orderings | p. 195 |
| Countable Abelian Groups | p. 199 |
| 01 and 01 Determinacy | p. 203 |
| The 01 and 01 Ramsey Theorems | p. 210 |
| Conclusions | p. 215 |
| $$ Comprehension | p. 217 |
| Perfect Kernels | p. 217 |
| Coanalytic Uniformization | p. 221 |
| Coanalytic Equivalence Relations | p. 225 |
| Countable Abelian Groups | p. 230 |
| 01 $$ Determinacy | p. 232 |
| The 02 Ramsey Theorem | p. 236 |
| Stronger Set Existence Axioms | p. 239 |
| Conclusions | p. 240 |
| Models of Subsystems of Z2 | |
| ß-Models | p. 243 |
| The Minimum ß-Model of $$-CA0 | p. 244 |
| Countable Coded ß-Models | p. 248 |
| A Set-Theoretic Interpretation of ATR0 | p. 258 |
| Constructible Sets and Absoluteness | p. 272 |
| Strong Comprehension Schemes | p. 286 |
| Strong Choice Schemes | p. 294 |
| ß-Model Reflection | p. 303 |
| Conclusions | p. 307 |
| -Models | p. 309 |
| -Models of RCA0 and ACA0 | p. 310 |
| Countable Coded -Models of WKL0 | p. 314 |
| Hyperarithmetical Sets | p. 322 |
| -Models of 11 Choice | p. 333 |
| -Model Reflection and Incompleteness | p. 342 |
| -Models of Strong Systems | p. 348 |
| Conclusions | p. 356 |
| Non--Models | p. 359 |
| The First Order Parts of RCA0 and ACA0 | p. 360 |
| The First Order Part of WKL0 | p. 365 |
| A Conservation Result for Hilbert's Program | p. 369 |
| Saturated Models | p. 379 |
| Gentzen-Style Proof Theory | p. 386 |
| Conclusions | p. 388 |
| |
| Additional Results | p. 391 |
| Measure Theory | p. 391 |
| Separable Banach Spaces | p. 396 |
| Countable Combinatorics | p. 399 |
| Reverse Mathematics for RCA0 | p. 405 |
| Conclusions | p. 407 |
| Bibliography | p. 409 |
| Index | p. 425 |
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