Comment by thaumasiotes 4 days ago
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.
Ok, I see your problem now. Tuples aren’t axiomatic components of set theory. Try proving they exist then you’ll see the problem too.