Comment by moomin
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.
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.
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.