The question I am struggling with is as follows:
Suppose that $\{G_\alpha\}$ is a countable collection of countable groups. Show that $\ast_{\alpha}G_\alpha$ is countable.
The definition of countable that I am using is that $G$ there is some injective function $G \rightarrow \mathbb{N}$.
I am aware that one can show the Cartesian product of countable groups is countable by constructing a matrix, and I have been trying to do something similar for this problem, however I can't seem to get anywhere.
There are at most $\;\aleph_0\cdot\aleph_0=\aleph_0\;$ finite words that can be formed out of the elements of each group $\;G_\alpha\;$ .
Every word in that free product is the concatenation of a finite number of words ( namely, from a finite subset of $\;\{G_\alpha\}$ ) .
Thus, we have at most $$\aleph_0\cdot\aleph_0=\aleph_0\;\;\text {words in}\;\;*_\alpha G_\alpha$$