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Following is the definition of Infinite Cartesian Products.

enter image description here https://en.wikipedia.org/wiki/Cartesian_product

For cartesian product, why do not we just use cartesian product and get "n-tuples" instead of "arbitrary(possibly infinite) indexed family" of function sets?

I am not sure but the reason might be that there is not any usual way to select from tuples unless a function used so functions used in first place. (Actually there is projection map definition in wiki article but not sure about it.) (This answer is also about n-tuple item access; Mathematical symbol to reference the i-th item in a tuple?)

So, is not there any way other than functions to access n-tuple items? So, what is the point actually?

Asaf Karagila
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lockedscope
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    What exactly is an "$n$-tuple" when $n$ is infinite? The point is that we actually want to think of ordinary finite tuples themselves as functions: a $2$-tuple, for example, is a function with domain ${0,1}$, whose value on $0$ is the $0$th coordinate and whose value on $1$ is the $1$st coordinate. – Noah Schweber Oct 06 '20 at 22:21
  • So, is it just because of that we cannot define an infinite n-tuple? – lockedscope Oct 06 '20 at 22:44
  • I wouldn't phrase it that way. Rather, I'd say that we haven't actually defined finite tuples in a sufficiently precise way. Try to define (say) a $3$-tuple in a precise way and I think you'll see the issue. – Noah Schweber Oct 06 '20 at 22:47
  • @NoahSchweber are not tuples necessarily ordered? So, i understand like this, they are distinct by order but when we are building/defining them, they do not have any order. – lockedscope Oct 06 '20 at 22:47
  • "Ordered" is a strong word, and isn't always applicable: sometimes we consider tuples indexed by sets which aren't ordered in any nice way (e.g. an $\mathbb{R}^2$-indexed tuple). Rather, a tuple has distinguished coordinates. This is really the crucial property: what a tuple does is answer questions of the form "What is your term with index $i$?," which is ... basically the same thing that a function does ("What is your value on input $i$?"). – Noah Schweber Oct 06 '20 at 22:49
  • This pragmatic approach - basically, that a tuple is what it does, and can be generalized accordingly - is developed further in category theory via the notion of universal properties. See here in particular. – Noah Schweber Oct 06 '20 at 22:53
  • @NoahSchweber So, we cannot set item at index i and could not read too, so what could we do with them? i am confused, have no sense in math but only in programming!?. – lockedscope Oct 06 '20 at 22:54
  • I don't understand your comment. A tuple is exactly a function: you feed it an index and it spits out a value. The language of "setting values" isn't really appropriate for math; in math, objects are static, not dynamic. (OK fine there are immutable objects in programming too, but in general there is a notion of "changing an object" in programming which isn't used in mathematics - my broader point is that programming-based ideas aren't always appropriate to math.) – Noah Schweber Oct 06 '20 at 22:58
  • (Re: static vs. dynamic definitions, see also the discussion here.) – Noah Schweber Oct 06 '20 at 23:00
  • @NoahSchweber thanks, you're right. – lockedscope Oct 06 '20 at 23:02
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    @lockedscope: There was a user that insisted that we can prove the axiom of choice because we can apply a Python/JS like approach and if we have a collection of sets, we can simply x.map((y)=>some element of y), "programming logic" can be misleading if you don't really think about the underlying math below the actual operations. Once you realise that the iteration is really just hiding a for loop, you can see that there will be some indexing and finite-to-infinite failure (because in a proof for or while get unfolded to their linear execution, in some sense). The same goes here. – Asaf Karagila Oct 06 '20 at 23:33
  • So, when we try programmatic/algorithmic approach, we cannot solve it infinitely and so we cannot prove or reach to some result. Therefore, programmatic approaches are invalid for infinities. – lockedscope Oct 07 '20 at 10:03

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