Approaching Infinity

List of Figures Preface PART I: THE NEED FOR A THEORY OF INFINITY 1. The Prevalence of the Infinite 1.1. The Concept of Infinity and the Infinite 1.2. The Infinite in Mathematics 1.3. The Infinite in Philosophy 1.4. The Infinite in the Physical World 1.5. The Infinite in Modern Physics 1.6. Controversies 2. Six Infinite Regresses 2.1. The Regress of Causes 2.2. The Regress of Reasons 2.3. The Regress of Forms 2.4. The Regress of Resemblances 2.5. The Regress of Temporal Series 2.6. The Regress of Truths 2.7. Conclusion 3. Seventeen Paradoxes of the Infinite 3.1. A Word about Paradoxes 3.2. The Arithmetic of Infinity 3.3. The Paradox of Geometric Points 3.4. Infinite Sums 3.5. Galileo's Paradox 3.6. Hilbert's Hotel 3.7. Gabriel's Horn 3.8. Smullyan's Infinite Rod 3.9. Zeno's Paradox 3.10. The Divided Stick 3.11. Thomson's Lamp 3.12. The Littlewood-Ross Banker 3.13. Benardete's Paradox 3.14. Laraudogoitia's Marbles 3.15. The Spaceship 3.16. The Saint Petersburg Paradox 3.17. The Martingale Betting System 3.18. The Delayed Heaven Paradox 3.19. Conclusion PART II: OLD THEORIES OF INFINITY 4. Impossible Infinite Series: Two False Accounts 4.1. 'An Infinite Series Cannot Be Completed by Successive Synthesis' 4.2. 'An Infinite Series of Preconditions Cannot Be Satisfied' 4.3. Conclusion 5. Actual and Potential Infinities 5.1. The Theory of Potential Infinity 5.2. Why Not Actual Infinities? 5.3. Infinite Divisibility 5.4. Infinite Time 5.5. Infinite Space 5.6. Infinitely Numerous Numbers 5.7. Infinitely Numerous Abstract Objects 5.8. Infinitely Numerous Physical Objects 5.9. Conclusion 6. The Cantorian Orthodoxy 6.1. The Importance of Georg Cantor 6.2. Sets 6.3. Cardinal Numbers 6.4. 'Greater', 'Less', and 'Equal' 6.5. Many Sets Are Equally Numerous 6.6. The Diagonalization Argument 6.7. Cantor's Theorem 6.8. The Paradoxes of Set Theory 6.9. Other Paradoxes of Infinity 6.10. Conclusion PART III: A NEW THEORY OF INFINITY AND RELATED MATTERS 7. Philosophical Preliminaries 7.1. Metapreliminaries 7.2. Phenomenal Conservatism 7.3. Synthetic A Priori Knowledge 7.4. Metaphysical Possibility 7.5. Possibility and Paradox 7.6. A Realist View of Mathematics 8. Sets 8.1. Sets Are Not Collections 8.2. Sets Are Not Defined by the Axioms 8.3. Many Regarded as One: The Foundational Sin? 8.4. The Significance of the Paradoxes 8.5. Are Numbers Sets? 8.6. Set Theory and the Laws of Arithmetic 9. Numbers 9.1. Cardinal Numbers as Properties 9.2. Frege's Objection 9.3. Arithmetical Operations 9.4. The Laws of Arithmetic 9.5. Zero 9.6. A Digression on Large Numbers 9.7. Magnitudes and Real Numbers 9.8. Indexing Uses of Numbers 9.9. Other Numbers 10. Infinity 10.1. Infinity Is Not a Number 10.2. Infinite Cardinalities 10.3. Infinite Extensive Magnitudes 10.4. Infinite Intensive Magnitudes 10.5. Some A Priori Physics 11. Space 11.1. Pointy Space Versus Gunky Space 11.2. The Unimaginability of Points 11.3. The Zero Argument 11.4. When Zero Is Not Mere Absence 11.5. The Paradox of Contact 11.6. The Problem of Division 11.7. The Dimensionality of Space Is Necessary 11.8. The Measure-Theoretic Objection 12. Some Paradoxes Mostly Resolved 12.1. The Arithmetic of Infinity 12.2. The Paradox of Geometric Points 12.3. Infinite Sums 12.4. Galileo's Paradox 12.5. Hilbert's Hotel 12.6. Gabriel's Horn 12.7. Smullyan's Infinite Rod 12.8. Zeno's Paradox 12.9. The Divided Stick 12.10. Thomson's Lamp 12.11. The Littlewood-Ross Banker 12.12. Benardete's Paradox 12.13. Laraudogoitia's Marbles 12.14. The Spaceship 12.15. The Saint Petersburg Paradox 12.16. The Martingale Betting System 12.17. The Delayed Heaven Paradox 12.18. Comment: Shallow and Deep Impossibilities 13. Assessing Infinite Regress Arguments 13.1. The Problem of Identifying Vicious Regresses 13.2. Viciousness through Metaphysical Impossibility 13.3. Viciousness through Implausibility 13.4. Viciousness through Explanatory Failure 13.5. Conclusion 14. Conclusion 14.1. Why Study Infinity? 14.2. Troubles with Traditional Approaches 14.3. A New Approach to Infinity 14.4. Some Controversial Views about Sets, Numbers, and Points 14.5. Solving the Paradoxes 14.6. For Further Reflection, or: What Is Wrong with this Book?