Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters.

The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes.

Written by one of the world's most talented philosophers Addresses a key metaphysical concept the nature of the infinite that has baffled thinkers since Aristotle Written in Michael Huemer's trademark informal and highly engaging style Directly addresses the questions whether actual infinities are possible Develops a truly original and distinctive view of the infinite

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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.

Inhalt
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. 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?