Approaching Infinity
by Michael Huemer

Publisher: Palgrave Macmillan, 2016.

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List of Figures xi
Preface xii
PART I   The Need for a Theory of Infinity
1   The Prevalence of the Infinite 3
2   Six Infinite Regresses 9
3   Seventeen Paradoxes of the Infinite 17
PART II   Old Theories of Infinity
4   Impossible Infinite Series: Two False Accounts 41
5   Actual and Potential Infinities 49
6   The Cantorian Orthodoxy 71
PART III   A New Theory of Infinity and Related Matters
7   Philosophical Preliminaries 93
8   Sets 108
9   Numbers 119
10   Infinity 143
11   Space 162
12   Some Paradoxes Mostly Resolved 176
13   Assessing Infinite Regress Arguments 229
14   Conclusion 247
References 261
Index 269


From ancient times, infinity has been steeped in paradox. According to one famous argument, nothing can ever move, because to move from one point to another, one must first travel half the distance, then half the remaining distance, and so on. Another puzzle asks us to imagine that a lamp that starts out off, then is turned on after half a minute, off after a quarter minute, and so on; at the end of one minute, is it on or off? Most observers think that the first of these infinite series can be completed, but the second cannot. Why? This book addresses this and many other puzzles about infinity, most of which have no generally accepted solutions. A new theory of the infinite is advanced, on which an infinite series is uncompletable when it requires something to possess an infinite natural, intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters.

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