Undirecting Membership in Models of ZFA

Bea Adam-Day, University of Leeds. Part of the Postgraduate Logic Seminar series

It is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation, we obtain the ErdÅ‘s–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops or multiple edges . I will show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $\aleph_0$- categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $\aleph_0$-categorical, but has infinitely many 1-types.