Is the ordered pair definition $(,):=\{,\{,\}\}$ a good definition?

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There is no one fixed way to define an ordered pair in terms of sets. It is also common to define an ordered pair as $(,):=\{,\{,\}\}$. One can prove that $\{x_1, \{x_1,y_1\}\} = \{x_2, \{x_2,y_2\}\} \iff x_1 = x_2 \text{ and }y_1=y_2 \label{1}\tag{$*$}.$

I've tried the proof, and the process is questionable: take simplest case when $$ x=y,\, u=v,\, \langle x,y\rangle=\langle u,v\rangle \iff \{,\{\}\}=\{u,\{u\}\},$$ to which $x=u,\{\}=\{u\}$ is a solution. As we can immediately see, also $x=\{u\},u=\{\}$ can be a solution as long as there exist a set such that $x=\{\{x\}\}$, i.e. if $x$ is a set containing a set containing itself.
Now searching the internet for an example of a set containing itself, I've found this Quora Q&A, where it states that in ZFC set theory there isn't a set containing itself (so I think it also means a set containing a set containing itself wouldn't exist). On the other hand, other non-ZFC set theories allows a set containing itself to exist (so I assume a set containing a set containing itself would also exist under such theory), so under this theory we cannot prove \eqref{1}.

On the contrary to Kuratowski's definition $\langle x,y\rangle=\{\{x\},\{x,y\}\}$, Wiener's definition $$ \langle x,y\rangle=\{\{\{x\},\emptyset\},\{\{y\}\} $$ is a good one because it don't rely on such a particular axiom of set theory. It works in set theories other than ZFC, even works in naive set theory.

Answer 1

As other commenters have pointed out, there are multiple possible viable set-theoretic representations of ordered pairs. But it seems like the OP's specific difficulty is in understanding why the convention of $\{x, \{x, y\}\}$ actually "works", so to speak.

I'd argue it as follows (I’m assuming a “pure” set theory, without “urelements”): The key is whether, for any instance of $\{x, \{x, y\}\}$, one can unambiguously tell which element is intended as the "first" and which element is the "second". This is indeed possible, because necessarily $x \in \{x, y\}$ but it cannot be true that $\{x, y\} \in x$ (that would violate the Axiom of Foundation). So we can tell which element is the "first", and then trivially we know which is "second" (I've assumed $x$ and $y$ are distinct -- a nearly identical argument pertains if $x = y$)

So $\{x, \{x, y\}\}$ seems like a viable convention for representing ordered pairs, but I think the Kuratowski convention is preferrable in that it's just clearer how it works (as evidenced by OP's perplexity). Also it's not totally obvious to me that $\{x, \{x, y\}\}$ would work for every case in non-well-founded set theories (lacking an Axiom of Foundation)

Answer 2

It can definitely be valuable to know multiple implementations of the same concept, and to keep track of what axioms are required for a given implementation. (For a particularly subtle example of this, see the notion of "flat pairing function" in the context of $\mathsf{NF}$.) However, your title and some of the body of the question (" the process is questionable") overstates the case; there is absolutely nothing wrong with using a definition which is successful in the context within which you are working.

So there's nothing mathematically wrong with what you've written above - the ordered pair $\{x,\{x,y\}\}$ is less "axiomatically flexible" than certain other implementations - but that doesn't mean that that notion is bad in any way.