In everyday conversation we use words and phrases that are, at best, ambiguous and vague. Yet, despite the fuzziness we somehow manage to communicate with each other and get things done.

A cool thing about mathematicians is that they work really hard to give words and phrases precise meaning. Today I learned how they give precision to the phrase *infinite set*. I think it’s mind-blowing.

Consider the set of positive integers, 1, 2, 3, 4, …

Clearly it is an infinite set. But wait! What does “infinite set” mean? That’s one of those ambiguous, vague phrases. So, how to define it? Let’s approach the problem by identifying a characteristic of infinite sets.

Notice that each positive integer can be mapped to an even integer by multiplying the positive integer by 2:

For every positive integer there is a corresponding even integer. There is a 1-for-1 correspondence between the two sets.

Therefore the two sets have the same number of members!

This result is surprising in view of the fact that the set of even integers is a proper subset of the set of positive integers (3, for example, is in the positive set but not the even set). We are accustomed to thinking of a set as being “larger” than any of its proper subsets, but here we are inescapably led to conclude that sometimes a set and a proper subset of that set may have the same number of members.

That is really unusual behavior!

When we examine the sets that exhibit this unusual behavior, we find that they are just the ones that we would intuitively call *infinite*.

DEFINITION: A set is infinite if and only if it is equivalent to a proper subset of itself.

Wow!