Comparing Anti-foundation Axioms by Comparing Identity Conditions for Sets

Abstract In non-well-founded set theory, which anti-foundation axiom is philosophically justified, BAFA, FAFA, SAFA, AFA, or some other one? In this paper, we investigate a general approach to answering this question: first, considering which identity condition for sets is justified; second, considering which anti-foundation axiom it justifies. Specifically, we study in detail two plausible identity conditions, $ \text{IC}_{1} $ and $ \text{IC}_{2} $: we show that $ \text{IC}_{2} $ justifies $ \text{FAFA}_{2} $ and $ \text{IC}_{1} $ justifies AFA, and argue for AFA by offering an argument for $ \text{IC}_{1} $.