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Table of Contents Preface ix Acknowledgments xi 1 Introduction 1 2 Language Paradoxes 15 2.1 Liar! Liar! 15 2.2 Self-Referential Paradoxes 19 2.3 Naming Numbers 26 3 Philosophical Conundrums 31 3.1 Ships, People, and Other Objects 31 3.2 Hangin’ with Zeno and Gödel 41 3.3 Bald Men, Heaps, and Vagueness 50 3.4 Knowing about Knowing 57 4 Infinity Puzzles 65 4.1 Sets and Sizes 66 4.2 Infinite Sets 69 4.3 Anything Larger? 76 4.4 Knowable and Unknowable 85 5 Computing Complexities 97 5.1 Some Easy Problems 98 5.2 Some Hard Problems 109 5.3 They’re All Connected 121 5.4 Almost Solving Hard Problems 129 5.5 Even Harder Problems 131 6 Computing Impossibilities 135 6.1 Algorithms, Computers, Machines, and Programs 136 6.2 To Halt or Not to Halt? 139 viii Table of Contents 6.3 They’re All Connected 146 6.4 A Hierarchy of the Unknown 152 6.5 Minds, Brains, and Computers 157 7 Scientific Limitations 161 7.1 Chaos and Cosmos 161 7.2 Quantum Mechanics 175 7.3 Relativity Theory 214 8 Metascientific Perplexities 235 8.1 Philosophical Limitations of Science 235 8.2 Science and Mathematics 252 8.3 The Origin of Reason 272 9 Mathematical Obstructions 297 9.1 Classical Limits 298 9.2 Galois Theory 304 9.3 Harder Than Halting 309 9.4 Logic 320 9.5 Axioms and Independence 331 10 Beyond Reason 339 10.1 Summing Up 339 10.2 Defining Reason 345 10.3 Peering Beyond 349 Notes 355 Bibliography 379 Index 393 ...

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