Comparing Anti-foundation Axioms by Comparing Identity Conditions for Sets

TL;DR

This paper explores the philosophical justification of various anti-foundation axioms in non-well-founded set theory by analyzing the identity conditions they support for sets.

Contribution

It introduces a general framework linking set identity conditions to anti-foundation axioms, focusing on two specific identity conditions.

Findings

Identifies key identity conditions for sets in non-well-founded theories.

Analyzes how different anti-foundation axioms correspond to these identity conditions.

Provides insights into the philosophical justification of anti-foundation axioms.

Abstract

In non-well-founded set theory, which anti-foundation axiom is philosophically justified, BAFA, FAFA, SAFA, AFA, or some other? In this paper, we investigate a general approach to answering this question: first, consider which identity condition for sets is justified; second, consider which anti-foundation axiom it justifies. Specifically, we study in detail two plausible identity conditions.