Defining product $\prod\limits_{i \in I} A_i$ for an arbitrary indexing set $I$ I just want to be sure that I understand this definition and its implications.
Let $\{A_i\}_{i \in I}$ be a family of sets. My understanding is: there isn't a problem with defining a finite or even a countably infinite Cartesian product as ordered tuples, $(a_1, \ldots, a_n)$ or $(a_1, a_2, \ldots )$, but there is no such thing as an "uncountable tuple." So, instead of defining such a tuple, we define a function $f: I \to \bigcup\limits_{i \in I} A_i$ such that $f(i) \in A_i$. Such a function might be "associated" with an uncountable tuple, though there doesn't exist any such sequence. In the case where $I$ is finite or countably infinite, this $f$ is "in effect" a tuple provided that we treat it of the form $(f(1), f(2), \ldots)$, so it's, in effect, equivalent up to isomorphism of sets.
Is this the right idea?
 A: Even in the case of a countably infinite product, we generally define the product as the set of choice functions, taking the index set to be the natural numbers.
It's just a question of "what's an indexed tuple, really"? Maybe it's something that's so clear that it doesn't need a more primitive definition. But generally we just define it as the set of choice functions, since that's simpler and removes any doubt. And if we're aspiring to formalize things completely in set theory, there is certainly no primitive notion and we must define it.
Note that even finite products can be defined this way... just with $I$ as a finite set. It's really just that the the ordered tuple isn't up to the job of defining infinite (even countably infinite) products. Recall the ordered pair is (usually) defined from set theory as $(a,b)=\{\{a\},\{a,b\}\}$, and then $A\times B:=\{(a,b): a\in A, b\in B\}$ and, say, $A\times B\times C:=(A\times B)\times C.$ This inductive definition only covers the finite cases, and is actually kind of clunky compared to just using choice functions on finite sets for finite products. But the big caveat preventing us from doing away with tuples entirely is that you need the ordered pair to define what a function is in set theory in the first place.
