Let s denote the "virtual set" (x:x is an element of x). Does there exist a set theory T (based on classical logic and not so far proved inconsistent) such that: (1) T allows the existence of all the sets in ZF as well as many infinite self-membered sets. (2) T allows s to exist as an actual set and provides an answer to the question "Is s an element of s?" without engendering any paradox.
Yes, there is such a set theory. It is ZF minus foundation plus Aczel's "anti-foundation axiom".
One reference for this system is a book-length set of lecture notes by Aczel, http://standish.stanford.edu/pdf/00000056.pdf
In this theory, the collection of all sets that contain themselves is nonempty (in fact, it is a proper class) and contains elements that are themselves infinite.
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