I will start with a somewhat provocative statement: The set of ALL positive integers has the same size as the set of EVEN positive integers. At first, this sentence seems obviously wrong - I mean, one set is a proper subset of the other. How can they have the same size? But it is true. And hiding behind this statement is a lot of very beautiful math. In this post, we’ll explore what that math is. but first, let’s break this down What do I mean by “same size”? The fancy word for size is cardinality. We’ll say that two sets have the same cardinality if there is a perfect one-to-one correspondence between the two sets. So obviously, two finite sets have the same cardinality if and only if they have the same number of elements. But with infinite sets, things get weird. Here is a perfect 1-1 correspondence between the positive integers and the set of ALL integers: just take each positive integer and multiply it by two.