How would you "count" $\omega^\omega$

Question

$\omega^\omega$ can be seen as the limit of $\omega^n$ which are all countable sets, and is thus countable. For the latter sets, there is an "easy" way list the elements out, but how would you do it for $\omega^\omega$? That is, what would be a bijection from the positive integers to $\omega^\omega$?

Answer 1

Define $f:\omega^\omega\to\Bbb{N}^+$ such that $$f(\omega^{n_0}\cdot a_0+\cdots+\omega^{n_k}\cdot a_k):=p_{n_0}^{a_0}p_{n_1}^{a_1}\cdots p_{n_k}^{a_k}.$$ This function is well-defined because of uniqueness of the Cantor normal form of ordinals, and it is 1-1 function because the fundamental theorem of arithmetic. Also, you can prove that this function is onto.