TITLE: An issue with the Kuratowski ordered pair definition
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: In this context everything, including natural numbers, functions, Hilbert spaces, etc., gets encoded as a set, just as in this case an ordered pair gets encoded as a set.
It then becomes tempting to say that everything is a set.  That statement is ok if you restrict everything to its proper context.  This way of encoding things serves some purposes but it's not necessarily sacred or the last word for all eternity.