Preface xiii Introduction: Invention or Discovery? 1(5) 1 The Finite Universe 6(26) 1 Finitism 6(4) 2 Continuity and Infinity 10(2) 3 Zeno's Paradoxes 12(10) 4 The Universe and the Absolutely Infinite 22(10) 2 Classes and Aristotelian Logic 32(24) 1 Aristotelian Logic 33(6) 2 Aristotle on Knowledge of Universals 39(8) 3 Nominalism and Extensionalism 47(3) 4 Toward an Algebra of Classes 50(4) 5 Classical Finitism in Retreat? 54(2) 3 Permutations, Combinations and Infinite Cardinalities 56(12) 1 Finite Permutations and Combinations 57(2) 2 Probabilities as Limits of Infinite Sequences 59(3) 3 Infinite Cardinalities 62(2) 4 The Finitist Response 64(4) 4 Numbering the Continuum 68(27) 1 The Algebraization of Geometry 69(15) 2 The Arithmetization of Analysis 84(4) 3 Toward Infinite Ordinal Numbers 88(2) 4 Conflict with Classical Finitism 90(5) 5 Cantor's Transfinite Paradise 95(23) 1 Sets and Cardinal Numbers 96(8) 2 Transfinite Ordinal Numbers 104(3) 3 Ordinal Numbers and Cantor's Continuum Hypothesis 107(4) 4 Order Types 111(3) 5 Set Theoretic Paradoxes 114(4) 6 Axiomatic Set Theory 118(20) 1 Axiomatization 118(3) 2 The ZF Axioms 121(13) 3 Transfinite Numbers in ZF 134(4) 7 Logical Objects and Logical Types 138(37) 1 Frege, Logic and Arithmetic 138(8) 2 Frege's Universe 146(8) 3 Iterative Sets and Simple Types 154(4) 4 Russell's Logicist Reduction 158(17) 8 Independence Results and the Universe of Sets 175(17) 1 Godel's Constructible Universe 175(4) 2 Cardinals and Ordinals in Models 179(4) 3 Inner Models 183(2) 4 Generic Sets 185(7) 9 Mathematical Structure - Construct and Reality 192(32) 1 Appeals to Mathematical Intuition 192(6) 2 Appeals to Mathematical Consequences 198(3) 3 Descriptive Set Theory 201(6) 4 Foundations and Superstructures 207(17) Further reading 224(3) Bibliography 227(6) Glossary of Symbols 233(2) Index 235