There is a old paradox, originally due to Montague (JSL 20 (2), 1955, p.140) that only requires the most minimalist set theoretic apparatus, viz., the existence of singletons. It does not seem to be very well known, so here it is.
Let . Intuitively, is the collection of all sets that only belong to well-founded sets. Clearly, is too big to be a set, but that is not the point. The point is that this can be shown in classical predicate logic using only existence of singletons:
We proceed by cases, and derive a contradiction in each.
- ; then only belongs to well-founded sets. Since exists and , the latter must be well-founded, so . But is equivalent to , so , which is impossible if (for then itself is in the intersection ).
- . Then there must be a such that: and . In particular , so there is a which belongs to both and . Now, since :
and since , we have contradicting the choice of .