Mathematics

In ZFC the primitive "membership" usually has the statement "x is an element of the set y". My question is "is the element 'x'" of a set ever not a set within ZFC?

Comments

There's no right answer.

Zermelo's original set theory allowed "urelements", i.e. entities in the universe which are members of sets but not themselves sets. Some modern writers use "ZFC" to refer to a descendant of Zermelo's theory allowing urelements. George Tourlakis is an example, in his two volume Lectures in Logic and Set Theory.

Some other writers (perhaps the majority) use "ZFC" to refer to the correponding theory of "pure" sets, where there are no urelements and the members of sets are themselves always other sets. Kenneth Kunen is just one example in his modern classic Set Theory.

If you are interested in set theory as a tool, then the first line is arguably the more natural one to take. If you are interested in set theory for its own sake, then for most purposes you might as well take the second line (because it seems to make no big difference to the sort of questions that most set theorists are interested in: for example ZFC-with-urelements is equiconsistent with ZFC-for-pure-sets).

To add a bit more, there are some interesting applications of urelements in set theory. Perhaps the most famous example is Quine's theory New Foundations. NF, as it is known, which does not permit urelements, remains something of a puzzle: It is not known if it is consistent. But NFU, which is just NF plus urelements, is known to be consistent if Peano arithmetic is. See the wikipedia entry on NF for more.

I seem to recall some interesting results due to Vann McGee about ZFU, as well, but they do not come immediately to mind.

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