Well, you’d need some model of index, for one. And I’m not sure how you’d construct that with uncountably many elements. Even ignoring that, a set containing n elements is different from a set containing n sets of one element each.
Not saying your formulation is wrong, just that there’s a fair amount of non-obvious work to get to the classic formulation.
No. We don't.
Let's note first of all that (a, b, c, a) is a tuple, not a Cartesian product.
Where are you getting your ideas from? What do you think a Cartesian product is? Give me a definition.
Well, it's true that tuples aren't defined by the ZF axioms. They do have a conventional definition, which you appear not to know.†
Natural numbers also aren't defined by the ZF axioms; why do you think that the definition of a tuple should involve them? You're likely to have a hard time finding a textbook that agrees.
I'm getting an overwhelming sense here that you don't know the meaning of the things you say. If you don't know what a Cartesian product is, why do you think it makes sense to talk about what you can and can't do with it?
Why do you think it makes sense to write {a b c a}, which has no meaning?
† For reference: by the standard convention, the tuple (a, b, c, a) is represented as the set {{{a, {a, b}}, {{a, {a, b}}, c}}, {{{a, {a, b}}, {{a, {a, b}}, c}, a}}. You might notice that no numbers appear anywhere. You might also notice that regardless of representation - and you're free to use other representations - it will never be a Cartesian product, because "tuple" and "Cartesian product" refer to different things.
> Even ignoring that, a set containing n elements is different from a set containing n sets of one element each.
Different in what way? A set containing n sets of one element each is a set containing n elements.
In ZF, everything that's an element of a set is itself a set, so unless what's bothering you is the idea that "a set containing n elements" might contain the empty set, or a set with two elements, I'm not seeing it.
> you’d need some model of index, for one. And I’m not sure how you’d construct that with uncountably many elements
Why would the number of elements matter? Set elements aren't indexed. What are you using your model of index for?
I really feel I should repeat the question I asked you to begin with: You say you need to convert a Cartesian product into a set containing "its elements". In your mind, what is a Cartesian product, before that conversion takes place?