I recently came across the following definition while reading on wikipedia:
Suppose $X$ is a countable set and $f:X\to\mathbb{R}$ is a real-valued function. Then we have that $\sum_{x\in X}f(x)$ is absolutely convergent if and only if: \begin{equation} \text{sup}\left\{\sum_{x\in A}|f(x)|:A\subseteq X,A\ \text{is finite}\right\}<\infty \end{equation}
Naturally my first thought to define absolute convergence of a sum over a countable set would be to take an arbitrary bijection $g:\mathbb{N}\to X$ and see whether $\sum_{n=1}^\infty f(g(n))$ absolutely converges. Is there an issue in doing so?
No issue at all. The "sup" definition being finite is equivalent to what you want to do. It merely avoids specifying a particular ordering of the terms in the series. And the "sup" definition is equivalent to what you want to do only because we know that the value of an absolutely convergent series is independent of the ordering of its terms.