Submission date: 8 August 2019

Abstract:

We answer some questions about graphs which are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership x∈y∈x. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

Mathematics Subject Classification: 03C62 (Primary) 03C13, 03E30, 03E65 (Secondary)

Keywords and phrases: Anti-Foundation, double-membership graph, Gaifman's Theorem, Hanf's Theorem, membership graph, non-well- founded sets, reducts of set theory.

Full text arXiv 1908.02708: pdf, ps.