List of Figures xi
Preface xii
Part I: The Need for a Theory of Infinity
1. The Prevalence of the Infinite 3
1.1. The Concept of Infinity and the Infinite 3
1.2. The Infinite in Mathematics 4
1.3. The Infinite in Philosophy 5
1.4. The Infinite in the Physical World 6
1.5. The Infinite in Modern Physics 7
1.6. Controversies 7
2. Six Infinite Regresses 9
2.1. The Regress of Causes 9
2.2. The Regress of Reasons 10
2.3. The Regress of Forms 11
2.4. The Regress of Resemblances 12
2.5. The Regress of Temporal Series 14
2.6. The Regress of Truths 15
2.7. Conclusion 16
3. Seventeen Paradoxes of the Infinite 17
3.1. A Word about Paradoxes 17
3.2. The Arithmetic of Infinity 17
3.3. The Paradox of Geometric Points 18
3.4. Infinite Sums 20
3.5. Galileo’s Paradox 21
3.6. Hilbert’s Hotel 22
3.7. Gabriel’s Horn 23
3.8. Smullyan’s Infinite Rod 24
3.9. Zeno’s Paradox 25
3.10. The Divided Stick 27
3.11. Thomson’s Lamp 28
3.12. The Littlewood-Ross Banker 29
3.13. Benardete’s Paradox 31
3.14. Laraudogoitia’s Marbles 32
3.15. The Spaceship 33
3.16. The Saint Petersburg Paradox 34
3.17. The Martingale Betting System 35
3.18. The Delayed Heaven Paradox 36
3.19. Conclusion 38
Part II: Old Theories of Infinity
4. Impossible Infinite Series: Two False Accounts 41
4.1. “An Infinite Series Cannot Be Completed by Successive Synthesis” 41
4.2. “An Infinite Series of Preconditions Cannot Be Satisfied” 44
4.3. Conclusion 47
5. Actual and Potential Infinities 49
5.1. The Theory of Potential Infinity 49
5.2. Why Not Actual Infinities? 51
5.3. Infinite Divisibility 52
5.4. Infinite Time 57
5.5. Infinite Space 57
5.6. Infinitely Numerous Numbers 65
5.7. Infinitely Numerous Abstract Objects 67
5.8. Infinitely Numerous Physical Objects 68
5.9. Conclusion 69
6. The Cantorian Orthodoxy 71
6.1. The Importance of Georg Cantor 71
6.2. Sets 71
6.3. Cardinal Numbers 73
6.4. “Greater", “Less”, and “Equal” 74
6.5. Many Sets Are Equally Numerous 75
6.6. The Diagonalization Argument 78
6.7. Cantor’s Theorem 80
6.8. The Paradoxes of Set Theory 82
6.9. Other Paradoxes of Infinity 85
6.10. Conclusion 89
Part III: A New Theory of Infinity and Related Matters
7. Philosophical Preliminaries 93
7.1. Metapreliminaries 93
7.2. Phenomenal Conservatism 95
7.3. Synthetic A Priori Knowledge 97
7.4. Metaphysical Possibility 100
7.5. Possibility and Paradox 105
7.6. A Realist View of Mathematics 106
8. Sets 108
8.1. Sets Are Not Collections 108
8.2. Sets Are Not Defined by the Axioms 110
8.3. Many Regarded as One: The Foundational Sin? 111
8.4. The Significance of the Paradoxes 113
8.5. Are Numbers Sets? 114
8.6. Set Theory and the Laws of Arithmetic 116
9. Numbers 119
9.1. Cardinal Numbers as Properties 119
9.2. Frege’s Objection 120
9.3. Arithmetical Operations 122
9.4. The Laws of Arithmetic 123
9.5. Zero 124
9.6. A Digression on Large Numbers 127
9.7. Magnitudes and Real Numbers 129
9.8. Indexing Uses of Numbers 137
9.9. Other Numbers 138
10. Infinity 143
10.1. Infinity Is Not a Number 143
10.2. Infinite Cardinalities 148
10.3. Infinite Extensive Magnitudes 150
10.4. Infinite Intensive Magnitudes 151
10.5. Some A Priori Physics 158
11. Space 162
11.1. Pointy Space Versus Gunky Space 162
11.2. The Unimaginability of Points 163
11.3. The Zero Argument 164
11.4. When Zero Is Not Mere Absence 165
11.5. The Paradox of Contact 168
11.6. The Problem of Division 170
11.7. The Dimensionality of Space Is Necessary 171
11.8. The Measure-Theoretic Objection 173
12. Some Paradoxes Mostly Resolved 176
12.1. The Arithmetic of Infinity 176
12.2. The Paradox of Geometric Points 176
12.3. Infinite Sums 178
12.4. Galileo’s Paradox 180
12.5. Hilbert’s Hotel 182
12.6. Gabriel’s Horn 183
12.7. Smullyan’s Infinite Rod 184
12.8. Zeno’s Paradox 185
12.9. The Divided Stick 188
12.10. Thomson’s Lamp 194
12.11. The Littlewood-Ross Banker 201
12.12. Benardete’s Paradox 207
12.13. Laraudogoitia’s Marbles 211
12.14. The Spaceship 211
12.15. The Saint Petersburg Paradox 213
12.16. The Martingale Betting System 218
12.17. The Delayed Heaven Paradox 220
12.18. Comment: Shallow and Deep Impossibilities 223
13. Assessing Infinite Regress Arguments 229
13.1. The Problem of Identifying Vicious Regresses 229
13.2. Viciousness through Metaphysical Impossibility 230
13.3. Viciousness through Implausibility 232
13.4. Viciousness through Explanatory Failure 236
13.5. Conclusion 245
14. Conclusion 247
14.1. Why Study Infinity? 247
14.2. Troubles with Traditional Approaches 248
14.3. A New Approach to Infinity 248
14.4. Some Controversial Views about Sets, Numbers, and Points 249
14.5. Solving the Paradoxes 250
14.6. For Further Reflection, or: What Is Wrong with this Book? 251
References 262