What is the infinite?

What does it mean, exactly, to say that a set is infinite? Can we provide precise mathematical definitions of the finite and the infinite?

What is the Infinite?

We have gotten ahead of ourselves, talking so much about the infinite, but without yet having a clear definition of what it means to be infinite. What actually is the infinite? What exactly does it mean to say that a given set is infinite? How are we to describe this sublime, elusive notion? We should want a crisp, clear account, one suitable for mathematics.

Please enjoy this new installment from The Book of Infinity, a series of vignettes on infinity with all my favorite puzzles and paradoxes, serialized over the past year.

In this installment, we discuss the problem of how we are to define the infinite.

Suppose I turn the tables and ask the dual question: What is the finite? You might think, how absurd, obviously we know all about the finite. Right? To my way of thinking, however, the two questions are equally hard, for they carve out the same boundary between the finite and the infinite. To define the finite is to define the infinite by complement; to have a clear criterion for one is to have a clear criterion for the other.

Meanwhile, it is not so easy and clear after all how to define either the finite or the infinite. In mathematical history there have been many proposals, and so let us explore a little the spectrum of concepts on offer, from Aristotle to Galileo to Dedekind, Frege, Stäckel, Tarski, and others.