Why do we accept Kuratowski's definition of ordered pairs?

Solution 1:

We accept this definition because it works, and it works very well.

However do note that almost nobody cares about the actual encoding of ordered pairs, and in most cases set theorists don't really care either. You just want a definition which satisfies the property which you have quoted.

I can't recall any substantial proof in set theory which actually refer to this definition of ordered pairs. Instead we just use the fact that there is some $\varphi(x,y,z)$ which is true if and only if $z$ represents the ordered pair $(x,y)$. Had I chose a different encoding, the proof would stay the same.

So why do we use it? Well, it tells us that set theory is strong enough to support an internal definition of ordered pairs which is another "thumbs up" in its favor as a foundational theory. But also we use it because it was there when we needed such definition, and once you have a working machine you don't really bother replacing the cogs which work perfectly.

Note that on its own, the object of an ordered pair is very uninteresting. So we don't really bother with finding a "better" definition, because nobody really cares about how you encode. We just want to know that it exists within our mathematical universe, and if our mathematical universe is based on set theory (in one way or another) then we want to know that we don't need an extra type in the world in order to have ordered pairs.

Solution 2:

Apart from the one you quote, the second property about ordered pairs that is needed for development of set theory is that $$ A\times B = \{ x \mid \exists a\in A, b\in B: x = (a,b) \} $$ must be a set whenever $A$ and $B$ are. Kuratowski pairs makes this easy to prove because $A\times B$ is a subset of the power set of the power set of $A \cup B$.

One might imagine other definitions of ordered pairs where one needed to appeal to the Axiom of Replacement in order to prove that $A\times B$ always exists. Being able to do it with the Power Set and Subsets alone can be seen as somewhat tidier.

Solution 3:

It doesn't matter what definition of ordered pair you use: {{a},{a,b}} is fine.. {{0,a},{1,b}} is fine

What's really important is that the category of sets has products and the universal property of a product that gives us the ability of pair and unpair and gives the equality relation you stated.

Solution 4:

Alternatively, you might enhance the language of set theory to include the notion of ordered pairs and add a few axioms accordingly. But then you'd have all kinds of mixed objects: Sets with pairs of a set and a pair of sets and whatnot as elements producing the weirdest combinations in the results of e.g. taking powersets or unions. This includes the necessity to treat pairs as urelements but of a "non-atomic" nature. All in all the theory might become a lot more wound-up without any real gain. Kuratowski allows us to both work with ordered pairs and work in a world where everything is a set. While "custom-types" makes the everiday mathematical work easier, the set-theoretical "monoculture" makes the foundation comfortably more trust-worthy.

All we need about Kuratowski's definition is indeed the fundamental property of ordered pairs as you mention it. The same applies for other constructions that can be performed with set theory: All we usually need about $\mathbb N$ is that it obeys the Peano axioms. Where applicable, we may work with $\mathbb N$ as a given set with opaque elements $0,1,2,\ldots$ and there is no need to accept definitions like $0:=\emptyset, 1:=\{\emptyset\}, 2:=\{\emptyset,\{\emptyset\}\}, \ldots, Sn:=\{n\}\cup n$ that put this on set-theoretic grounds. Just as with pairs, there are alternatives (such as $Sn:=\{n\}$). It may not be of practical use, but is comforting that a set that models the Peano axioms can be constructed easily and need not be added explictly to the theory.

The further constructions of $\mathbb Z, \mathbb Q, \mathbb R, \mathbb C$ may be used as further examples: We don't work with (Kuratowski) pairs of sets of equivalence classes of pairs of equivalence classes of pairs of finite ordinals when computing zeroes of the Riemann zeta function, rather we work with facts like that $\mathbb C$ is an algebraically closed field and do as if e.g. complex numbers are just objects of a new type. And again there are differnt choices that don't make a difference in the end (real numbers as Dedekind cuts or as equivalence classes of Cauchy ssequences?). A need to look under the hood of all constructions involved may arise when we want to communicate unambiguously what we talk about ("Why is $0.\bar9=1$?", though that could be done with the axioms the constructions model) and want to face doubts about the "existence" of the subject matter ("Why can we talk about the set of all Lebesgue integrable functions but not about the set of all sets?").

Solution 5:

We all agree about the "arbitrary" nature of Kuratowski's definition, and about the possibility of replacing it with other "conventions" (equally arbitrary), provided that they satisfy the basic properties (already discussed).

But please, note that the lack of a mathematical definition of ordered couple forced two of the "founding fathers" of modern mathematical logic to "waste" hundreds of pages (full of formulae) of their masterpiece (see Alfred North Whitehead & Bertrand Russell, Principia Mathematica Vol I (1910) : SECTION C: CLASSES AND RELATIONS, strating from :§20. GENERAL THEORY OF CLASSES [page 196-on] and §21. GENERAL THEORY OF RELATIONS [page 212-on]) in order to "duplicate" all the basic result stated in terms of propositional functions of one variables

$\psi(x)$

"representing" classes,

also for propositional functions of two variables

$\psi(x, y)$,

"representing" (dyadic) relations.

Thus, all we "mathematical students" must every night address a little prayer for the benefit of Kuratowski's soul ...