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I am not formally educated mathematically, but still, I have a question that popped into my mind and I would really appreciate it if someone could help me understand it.

from my very small knowledge about the subject, I have an understanding that unordered sets are used as a foundation for all of mathematics, meaning that any mathematical structure should possibly be reduced into an unordered set.

So can anyone explain to me with no technical terms and for someone with no background knowledge in the subject how could an ordered set like this one (1,2,3) be represented using something that fundamentally has no meaning for an order?

I found this answer that suggests that idea of representing this is based upon the assumption that

the ordered set "a"

a = (a)

If that's indeed correct

I find it a very strange assumption because we don't assume for example that

a = {a}

like what is the justification/explanation for such an assumption?

Thanks a lot in advance.

A. S.
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    See also How can an ordered pair be expressed as a set? – JMoravitz Mar 07 '23 at 13:10
  • @JMoravitz thanks a lot – A. S. Mar 07 '23 at 13:11
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    With regards to triples and other $n$-tuples which are larger, there are ways to extend these common definitions. For instance, we might formally treat $(a,b,c)$ as $((a,b),c)$ taking the elements two at a time and using the kuratowski definition here twice, ${{{{a},{a,b}}},{{{a},{a,b}},c}}$. In the end, the answer of the first linked duplicate says it best. "Which definition we pick is not really important. What is important is that the objects we choose to represent ordered pairs must behave like ordered pairs. If we get that much, we are mathematically satisfied." – JMoravitz Mar 07 '23 at 13:13
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    (hopefully I used the correct number of braces in that example. It gets rather messy, but hopefully you get the idea). For that reason, the Hausdorf definition is considerably easier to read. $(a,b,c)$ is simply ${{a,1},{b,2},{c,3}}$ – JMoravitz Mar 07 '23 at 13:15
  • @JMoravitz I really appreciate your help, thank you – A. S. Mar 07 '23 at 13:25
  • @JMoravitz may I ask, how come in the first method there is no difference between (x,y,z) and ((x,y), z), while in the second method, there is clearly a difference, for example, if we first construct (a,b) first and then substitute its occurrences in ((a,b),c) we get {{{a,1}, {b,2}}, 1}, {c, 2}}, but if we do it with (a,b,c) we end up with {{a,1}, {b,2}, {c,3}} which are clearly two different results, so is (a,b,c) really equal to ((a,b),c)? it's a little inconsistent isn't it? – A. S. Mar 07 '23 at 16:42
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    If you are in a situation where you want to emphasize that they are different, then you use rules and definitions that result in that being the case. The point is that we get to choose the rules we work with in order to get the outcome we desire. We know how we want triples to behave when comparing to other triples. If you want to be able to compare triples to pairs as well, then you'll have to be more careful about how you word it potentially. I refer you again to the quote I quoted earlier. – JMoravitz Mar 07 '23 at 16:47
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    The problems we wish to solve and answers we want to get come first. The definitions come after. We create the definitions in such a way that they are useful to us for solving what it is we wanted to solve and so they have the behavior we desired. Take numbers for instance... the human race and society have intuitively used numbers like $1,2,3,4,\dots$ for tens of thousands of years. It wasn't until relatively recently that we decided to define them by way of $0,1,2,\dots$ being ${},{{}},{{{}},{}},\dots$ respectively. – JMoravitz Mar 07 '23 at 16:51

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