  When Cantor applied this conception of cardinality to infinite sets, he came to the initially surprising conclusion that there are different kinds of infinity. There are relatively small infinite sets like the set of even numbers, the set of integers, or the set of rational numbers. These sets can all be put into one-to-one correspondence with the natural numbers; they are called countably infinite. In contrast, there are much “larger” infinite sets like the set of real numbers, the set of complex numbers, or the set of all subsets of the natural numbers. These sets are too big to be put into one-to-one correspondence with the natural numbers; they are called uncountably infinite. Cantor's Theorem, then, is just the claim that there are uncountably infinite sets—sets which are, as it were, too big to count as  countable.[2]