Partition of $\mathbb{N}$ into $\aleph_0$ infinite sets Theorem - There is a division of $\mathbb{N}$ for $\aleph_0$ sets so every set has cardinal of $\aleph_0$.
I don't know even where to start in order to prove this theorem. 

For example:
Let a natural number $a$ belong to the $n$th set of the partition if $a$ has exactly $n$ prime factors. 
(And put $0$ and $1$ to any of them, say to the first one.)
