Preface to the Paperback Edition ix
Preface xi
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
lnfinitesimals 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
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
Godel's incompleteness Theorem 157
Conversations with Godel 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
Satori 214
Puzzles and Paradoxes 219
Excursion One: The Transfinite Cardinals 221
On and Alef-One 221
Cardinality 226
The Continuum 238
Large Cardinals 253
Excursion Two: Godel's Incompleteness Theorems 267
Formal Systems 267
Self-Reference 280
Godel's Proof 285
A Technical Note on Man-Machine Equivalence 292
Answers to the Puzzles and Paradoxes 295
Notes 307
Bibliography 329
Index 339
Table of Contents
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