What is the advantage of using an indexing set?

Question

For countable sets, what is the advantage of using an indexing set, such as $i \in I$, compared to just using the naturals and the normal enumeration of $1, 2, 3, \ldots$? To me it seems they equivalent unless the sets are uncountable, in which you cannot use something like $\mathbb{N}$ to index your set.

Answer 1

Even ignoring the advantage of avoiding a notational disagreement between countable and uncountable cases, just because something is countable doesn't mean that there is a canonical bijection between it and $\mathbb{N}$. By being flexible about index sets we can make sure that the Cartesian products we build don't have (or require us to input any!) any "extraneous information."

To see how this might get used in an argument, consider for example the following:

$(*)\quad$ Given a countable set $S\subseteq\mathbb{R}$, let $\mathcal{F}$ be all the ways of assigning to each $s\in S$ some $t<s$.

For each $S$ this is a Cartesian product; specifically, it's $$\prod_{s\in S}\{t\in\mathbb{R}: t<s\}.$$ Now if we really want to, since each $S$ is countable we could instead look at $$\prod_{n\in\mathbb{N}}\{t\in\mathbb{R}: t<f(n)\}$$ for some fixed bijection $f:\mathbb{N}\rightarrow S$, but this adds a layer of complexity: we need to fold in some choice of $f$, and in general there won't be a "best" choice to make here. In particular, if we insist on doing this then the definition of $\mathcal{F}$ given in $(*)$ is incomplete for our purposes since it doesn't tell us which $f$ to pick.