Prove that $\{X \in P(Z)| X \text{ is finite}\}$ is enumerable.

Question

I am not sure how to approach this problem. if you could help it would be great.

Answer 1

Let $F_n = \{-n,-n+1,...,0,...,n-1, n \}$. Then $|{\cal P}(F_n)| = 2^{2n+1}$, hence finite, of course. We have ${\cal P}(F_n) \subset {\cal P}(F_{n+1})$, so the power sets are nested.

Note that $\{X \in {\cal P}(\mathbb{Z}) | X \text{ is finite} \} = \cup_{n=1}^\infty {\cal P}(F_n) = {\cal P}(F_1) \cup ( \cup _{n=2}^\infty ( {\cal P}(F_n) \setminus {\cal P}(F_{n-1})))$.