In Infinity and the Mind Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations.

Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.

Infinity and the Mind can be read and enjoyed by experts and nonexperts alike. Rudy Rucker is a talented logician who draws on his talents as a science-fiction writer and cartoonist to convey his ideas. This makes for not only a solid, accurate, and informative book but also a good read.

Rudy Rucker, set theorist and science-fiction author, has continued the tradition ... of making mathematics and computer science accessible to the intellectually minded layperson.... Infinity and the Mind is funny, provocative, entertaining, and profound.

Attempts to put Gödel's theorems into sharper focus, or at least to explain them to the nonspecialist, abound. My personal favorite is Rudy Rucker's Infinity and the Mind which I recommend without reservation.

Rudy Rucker is a talented logician who draws on his skills as a science-fiction writer and cartoonist to convey his ideas. This makes for not only a solid, accurate, and informative book but also a good read.

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