Abstract

The Cantorian set theory survived the Russell Paradox by means of axiomatizing the theory into the standard theory named ZF. ZF has a set theoretical counterpart named FA (Foundation Axiom) to the vicious-circle principle of Russell's ramified type theory. Despite the principle, circular objects and phenomena are ubiquitous in many applications fields of ZF. For modeling such circular things directly as circular sets, Aczel replaced FA with his Anti-Foundation Axiom (AFA) to allow non-well-founded sets in a strong way. The foundation of the new set theory is explained in details.