# An issue with the Kuratowski ordered pair definition

I recently came across the Kuratowski definition of ordered pairs (https://en.wikipedia.org/wiki/Ordered_pair#Defining_the_ordered_pair_using_set_theory).

I'm a bit confused about how the coordinates can be extracted. According to Wikipedia, if we have an ordered pair $p = (a,b) = \{\{a\}, \{a,b\}\}$, we can extract the first coordinate with $a = \bigcup \bigcap{p}$ where $\bigcup$ and $\bigcap$ are arbitrary unions and intersections respectively.

According to the definition of an arbitrary intersection, $\bigcap(\{\{a\}, \{a,b\}\}) = \{a\}$. $\bigcup\{a\}$ however seems a bit problematic. For the definition of an arbitrary union to work $a$ would have to be a set. What if $a$ is e.g. a natural number? A similar issue occurs with extracting the second pair.

Perhaps I'm being overly pedantic or does the definition just automatically assume that $a$ and $b$ are sets?

• This definition of ordered pair is used in the context of a set theory like $\mathsf{ZF(C)}$, in which everything is a set. – Brian M. Scott Oct 23 '15 at 22:18

This is not problematic when you remember that in modern set theory, everything is a set (even the natural numbers!). In particular $a$ is a set.
So $\bigcup\{a\}$ makes sense again, and it is not hard to see why you get $a$.
If you want to take some things as non-sets, then you need to define some class-functions $\varphi_1(z,x)$ and $\varphi_2(z,y)$ which essentially say that $z$ looks like a Kuratowski ordered pair, and $x$ is the unique element which appears in $\bigcap z$; and $\varphi_2$ would say a similar thing about how you extract $y$ from $z$.