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

  • Thread starter bentley4
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  • #1
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Main Question or Discussion Point

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?)
 

Answers and Replies

  • #2
Deveno
Science Advisor
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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.
 
  • #3
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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.)
 
  • #4
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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.
 

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