Kripke-Platek Set Theory and the Anti-Foundation Axiom

The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in diiering elds such as mathematical logic, computer science, artiicial intelligence, linguistics, cognitive science, and philosophy. Logicians rst explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. 6], 2]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. 4]). Instead of the Foundation Axiom these set theories adopt the so-called Anti