TITLE: Why do we accept Kuratowski's definition of ordered pairs?
QUESTION [19 upvotes]: I've been struggling understanding Kuratowski's definition of ordered pairs.
I understand what it means but I don't see why I should accept it.
I've seen this question and this one, most importantly --- through reading the wiki page I've realised one thing.
The only reason $(a,b)=\{\{a\},\{a,b\}\}$ is accepted is because it satisfies

$(a,b)=(c,d) \iff (a=c) \land (b=d)$

Am I not misunderstanding this?
If I can come with my own exotic definition which satisfies the above iff statement, would it be accepted?

REPLY [6 votes]: 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 ...