Russell's paradox

Russell's paradox is a famous paradox arising in elementary set theory. In simplest terms, it points out the logical inconsistencies that arise from allowing a set to be considered a member of itself.

One illustration of the paradox is the following:
 * Consider a bunch of libraries in a large city, each of which contains a catalog (in book form) listing all books owned by the library.
 * Some libraries list the catalog book itself in the catalog, since it is also a book they own.
 * Some libraries don't list the catalog itself in the catalog, thinking it unnecessary.
 * Now consider two master catalogs to be held at the city's Central Library:
 * the first lists all catalogs that contain themselves;
 * the second lists all catalogs that don't contain themselves.
 * Clearly the first catalog can list itself; that's no problem. But should the second catalog not list itself? Either way one answers this question leads to a contradiction:
 * If the second catalog lists itself, then it clearly doesn't belong in the list of catalogs that don't list themselves.
 * But if the second catalog doesn't list itself, then it does belong in the list of all catalogs that don't list themselves.

The paradox can be avoided (in this case) either by explicitly ruling out the possibility of a catalog listing itself (in which case there's no need to specify that the catalogs "don't list themselves") or by defining a "catalog of books owned by the library" as a different kind of thing than the "books owned by the library". See Zermelo–Fraenkel set theory and Von Neumann–Bernays–Gödel set theory for how Russell's paradox is overcome in formal set theory.

The paradox in apologetics
The Kalam cosmological argument can be seen as being vulnerable to Russell's paradox when it considers the universe, which might be defined as the set of "all things that have come into existence", as being something that has come into existence. See the Kalam article for details.