Princeton University Press | 2005 | 368 páginas | rar - epub | 8,4 Mb
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. 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.
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.
Contents
Preface to the 2005 Edition ix
Preface to the Paperback Edition xvii
Preface xix
Chapter One: Infinity 1
A Short History of Infinity
Physical Infinities;
Temporal Infinities; Spatial Infinities; Infinities in the Small; Conclusion
Physical Infinities;
Temporal Infinities; Spatial Infinities; Infinities in the Small; Conclusion
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; The Alefs
Infinitesimals and Surreal Numbers 78
Higher Physical Infinities 87
Puzzles and Paradoxes 91
Chapter Three: The Unnameable 93
The Berry Paradox 93
Naming Numbers; Understanding Names
Random Reals 107
Constructing Reals; The Library of Babel ; Richard’s Paradox; Coding the World
What is Truth? 143
Conclusion 152
Puzzles and Paradoxes 155
Chapter Four: Robots and Souls 157
Gödel’s Incompleteness Theorem 157
Conversations with Gödel 164
Towards Robot Consciousness171
Formal Systems and Machines; The Liar Paradox and the Non-Mechanizability of Mathematics; Artificial Intelligence via Evolutionary Processes; Robot Consciousness
Beyond Mechanism?185
Puzzles and Paradoxes187
Chapter Five: The One and the Many189
The Classical One/Many Problem189
What is a Set?191
The Universe of Set Theory196
Pure Sets and the Physical Universe; Proper Classes and Metaphysical Absolutes
Interface Enlightenment206
One/Many in Logic and Set Theory; Mysticism and Rationality; Satori
Puzzles and Paradoxes219
Excursion One: The Transfinite Cardinals 221
On and Alef-One 221
Cardinality 226
The Continuum 238
Large Cardinals 253
Excursion Two: Gödel’s Incompleteness Theorems 267
Formal Systems 267
Self-Reference 280
Gödel’s Proof 285
A Technical Note on Man-Machine Equivalence 292
Answers to the Puzzles and Paradoxes 295
Notes 307
Bibliography 329
Chapter Two: All the Numbers 53
From Pythagoreanism to Cantorism 53
Transfinite Numbers 64
From Omega to Epsilon-Zero; The Alefs
Infinitesimals and Surreal Numbers 78
Higher Physical Infinities 87
Puzzles and Paradoxes 91
Chapter Three: The Unnameable 93
The Berry Paradox 93
Naming Numbers; Understanding Names
Random Reals 107
Constructing Reals; The Library of Babel ; Richard’s Paradox; Coding the World
What is Truth? 143
Conclusion 152
Puzzles and Paradoxes 155
Chapter Four: Robots and Souls 157
Gödel’s Incompleteness Theorem 157
Conversations with Gödel 164
Towards Robot Consciousness171
Formal Systems and Machines; The Liar Paradox and the Non-Mechanizability of Mathematics; Artificial Intelligence via Evolutionary Processes; Robot Consciousness
Beyond Mechanism?185
Puzzles and Paradoxes187
Chapter Five: The One and the Many189
The Classical One/Many Problem189
What is a Set?191
The Universe of Set Theory196
Pure Sets and the Physical Universe; Proper Classes and Metaphysical Absolutes
Interface Enlightenment206
One/Many in Logic and Set Theory; Mysticism and Rationality; Satori
Puzzles and Paradoxes219
Excursion One: The Transfinite Cardinals 221
On and Alef-One 221
Cardinality 226
The Continuum 238
Large Cardinals 253
Excursion Two: Gödel’s Incompleteness Theorems 267
Formal Systems 267
Self-Reference 280
Gödel’s Proof 285
A Technical Note on Man-Machine Equivalence 292
Answers to the Puzzles and Paradoxes 295
Notes 307
Bibliography 329
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