Approaching Infinity

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