According to wikipedia, urelements (objects that are elements of a set but which are not sets themselves) do not arise in Zermelo Frankel set theory. The following operations in ZF seem to me to generate urelements though.
Let Inf be a set with a countably infinite number of elements. The existence of such a set is entailed by the Empty Set Axiom and the Axiom of Infinity.
Let Re be the powerset of Inf. The Power Set Axiom entails that Re is a set, because Inf is.
But what about the elements of Re? Any element in Re that can be described by a finite formula in the language of ZF is a set, because we can use that formula in the Axiom Schema of Separation (aka Specification) to isolate that element and assert that it is a set. But only countably many elements can be so isolated and given set status. We know that the cardinality of Re is the same as that of the reals, so there will be uncountably many other elements of Re for which there is no way they can be 'turned into sets' via the ZF axioms. Or at least, I can't currently see a way to do so.
If there is indeed no way to do this then all those other elements of Re are not sets, and hence are urelements, contrary to the claim in wikipedia.
Is there a way to resolve this difficulty?
Thanks in anticipation.