Approaching Infinity

Michael Huemer received a B.A. from UC Berkeley and a Ph.D. from Rutgers University. He is presently a full professor at the University of Colorado, where he has taught since 1998. He has published three single-author scholarly books, one edited anthology, and more than fifty academic articles in epistemology, ethics, political philosophy, and metaphysics. His articles have appeared in such journals as the Philosophical Review, Mind, the Journal of Philosophy, Ethics, and others. Michael's first book, Skepticism and the Veil of Perception, significantly advanced the theory of Phenomenal Conservatism in epistemology, which is now considered one of the leading theories of justified belief and is the focus of the recent anthology, Seemings and Justification (Oxford, 2013). His second book, Ethical Intuitionism, is one of the leading contemporary defenses of ethical intuitionism and of moral realism more generally. It has been assigned as course reading by two colleagues at my own university, in addition to philosophers at the University of Massachusetts at Amherst, Lafayette College, Huron University College, Syracuse University, and Princeton University. It was the subject of a book symposium in Philosophy and Phenomenological Research. Michael's most recent book, The Problem of Political Authority, was published in 2013.

• List of Figures PrefacePART I: THE NEED FOR A THEORY OF INFINITY1. 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. ConclusionPART II: OLD THEORIES OF INFINITY4. 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 Numerous6.6. The Diagonalization Argument 6.7. Cantor's Theorem 6.8. The Paradoxes of Set Theory 6.9. Other Paradoxes of Infinity 6.10. ConclusionPART III: A NEW THEORY OF INFINITY AND RELATED MATTERS7. 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. OtherNumbers 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?

“Huemer writes books that are well-argued and thought-provoking; it seems that each reader has to find for him/herself how far this variegated discussion leads to an improvement of the current state of affairs.” (Michel Weber, zbMATH 1426.03004, 2020)“Michael Huemer’s new book Approaching Infinity squarely takes on problems related to understanding the infinite. … Huemer’s book is indeed a very good contribution to the study of the infinite. … Huemer’s book gives a challenging and well thought out account of the infinite. His presentation is one that will surely engage anyone interested in this topic. … Approaching Infinity should be required reading for anyone with an interest in this fascinating topic.” (Biagio Gerard Tassone, dialectica, Vol. 71 (2), June, 2017)“Approaching Infinity is entertaining and competently written. It is recommended to anybodywho finds the puzzles of the infinite intrinsically interesting, but also to those who think that the paradoxes of the infinite were totally tamed by Aristotle or contemporary mathematics.” (Christopher M.P. Tomaszewski, The Review of Metaphysics, Vol. 71 (3), 2017)