TABLE OF CONTENTS Preface to the 2005 Edition ix Preface to the Paperback Edition XVII Preface xix Chapter One: Infinity 1 A Short History of Infinity 1 Physical Infinities 9 Temporal Infinities 10 Spatial Infinities 15 Infinities in the Small 24 Conclusion 34 Infinities in the Mindscape 35 The Absolute Infinite 44 Connections 49 Puzzles and Paradoxes 51 Chapter Two: All the Numbers 53 From Pythagoreanism to Cantorism 53 Transfinite Numbers 64 From Omega to Epsilon-Zero 65 The Alefs 73 Infinitesimals and Surreal Numbers 78 Higher Physical Infinities 87 Puzzles and Paradoxes 91 Chapter Three: The Unnameable 93 The Berry Paradox 93 Naming Numbers 95 Understanding Names 100 vi TABLE OF CONTENTS Random Reals 107 Constructing Reals 108 The Library of Babel 120 Richard's Paradox 126 Coding the World 130 What is Truth? 143 Conclusion 152 Puzzles and Paradoxes 155 Chapter Four: Robots and Souls 157 Gooe!'s Incompleteness Theorem 157 Conversations with Gooe! 164 Towards Robot Consciousness 171 Formal Systems and Machines 172 The Liar Paradox and the Non-Mechanizability of Mathematics 175 Artificial Intelligence via Evolutionary Processes 180 Robot Consciousness 183 Beyond Mechanism? 185 Puzzles and Paradoxes 187 Chapter Five: The One and the Many 189 The Classical One/Many Problem 189 What is a Set? 191 The Universe of Set Theory 196 Pure Sets and the Physical Universe 196 Proper Classes and Metaphysical Absolutes 202 Interface Enlightenment 206 One/Many in Logic and Set Theory 207 Mysticism and Rationality 209 Satod 214 Puzzles and Paradoxes 219 Excursion One: The Transfinite Cardinals 221 On and Alef-One 221 Cardinality 226 The Continuum 238 Large Cardinals 253 TABLE OF CONTENTS vii Excursion Two: Godel's Incompleteness Theorems 267 Formal Systems 267 Self-Reference 280 Godel's Proof 285 ATechnical Note on Man-Machine Equivalence 292 Answers to the Puzzles and Paradoxes 295 ~~ ~7 Bibliography 329 Index 339 .