Do tuples exist which aren't elements of a cartesian product of sets?

[bentley4]

Do tuples exist which aren't elements of a cartesian product of sets?
Can you just write an ordered list of elements which does not necessarily have to be defined in sets? (or does every tuple need to be defined through sets in order for it to rigourously exist in mathematics?)

 

[Deveno]

it's not clear what you're asking.

normally, an ordered tuple is a subset of SOME set AxBxC, but the sets involved might be unusual.

 

[alexfloo]

The answer is yes.

Suppose you give me the triple (a, b, c). Then this is an element (in fact, the only element) of {a}x{b}x{c}.

This works for all (finite) tuples. (For infinite tuples, which are actually just sequences, we need to assume the axiom of choice to guarantee that there such a set exists.)

 

[eztum]

Can you just write an ordered list of elements which does not necessarily have to be defined in sets? (or does every tuple need to be defined through sets in order for it to rigourously exist in mathematics?)

An natural question, particularly for a mathematically interested programmer: Wheras defining arrays (i.e. tuples) of any length is commonplace in all sensible programming languages, sets are absent from wide spread languages such as C.
Actually one can build the foundations of mathematics in a way that tuples are closer to the grounds than sets.
Most professors in mathematics probably feel that it would create more trouble than benefits if one would deviate from the narrow set based presentation of mathematics in courses.
Others feel that the strengthening role of computers, computing, and computation in science asks for a redesign of the taught foundations of mathematics.

You are on a good way; keep your eyes open an look behind the omnipresent orthodoxy.

 

[blue_raver22]

I can confirm that tuples do not necessarily have to be defined in sets in order to exist. A tuple is simply a finite ordered list of elements, and it can exist independently of any set. While tuples are often used in the context of set theory and cartesian products, they can also exist in other mathematical structures such as vector spaces and matrices. Therefore, it is possible for tuples to exist that are not elements of a cartesian product of sets. However, in order for a tuple to be rigorously defined in mathematics, it must have well-defined elements and a specific order. This can be achieved through sets, but it is not a requirement.