What is the free abelian group on $\mathbb{N}$?

Question

I learnt that the free abelian group on a set $X$ is the group $(\operatorname{Hom}(X, \mathbb{Z}), +)$. Okay, this sounds all right, but I also know the famous result that $\mathbb{Z}^{\mathbb{N}}$ is not free abelian. Since $\mathbb{Z}^{\mathbb{N}}=\operatorname{Hom}({\mathbb{N}, \mathbb{Z}})$, I am pretty confused and I have two questions:

1. Are the concepts of the "free abelian group on some set" and that of "free abelian group" (this is the one where we take the definition as that of a free $\mathbb{Z}$-module) two different things?
2. What is the free abelian group on $\mathbb{N}$, since it doesn't seem that $\operatorname{Hom}({\mathbb{N}, \mathbb{Z}})$ is what we are looking for?

Answer 1

The free Abelian group on a set $X$ is not given by the set of functions from $X$ to $\mathbb{Z}$. It is given by the subset of those functions that take a nonzero value on only finitely many inputs. If you re-do your investigations with that change, everything should make a lot more sense.