  The term non-wellfounded set refers to sets which contain themselves as members, and more generally which are part of an infinite sequence of sets each term of which is an element of the preceding set.  So they exhibit object circularity in a blatant way. Discussion of such sets is very old in the history of set theory, but non-wellfounded sets are ruled out of Zermelo-Fraenkel set theory (the standard theory) due to the Foundation Axiom (FA).  As it happens, there are alternatives to this axiom FA.  This entry is especially concerned with one of them, an axiom first formulated by Marco Forti and Furio Honsell in a 1983 paper.  It is now standard to call this principle the Anti-Foundation Axiom (AFA), following its treatment in an influential book written by Peter Aczel in 1988.