here's one question that's bugging me nuts.

Cantor defined different kinds of infinity, named aleph_0, aleph_1, ..., each one the size of the powerset of a set of the previous size. The naturals, and therefore all infinite subsets of it, are countably-infinite with cardinality aleph_0.

Take now the set of primes. It's an infinite subset of the naturals. Now take the powerset of the primes, and it's not difficult to see a one-to-one correspondence to the set of squarefree numbers: the unique factorization of each squarefree number corresponds to a subset of the primes.

But the squarefree are also a subset of the naturals. So what gives. (Likely it's my understanding of Cantor's theory that is flaky.)

Thanks!