Undirecting membership in models of ZFA

TL;DR

This paper explores how removing the Axiom of Foundation from set theory affects the structure of graphs formed from set membership, revealing diverse properties like randomness, categoricity, and component structure.

Contribution

It analyzes the properties of graphs derived from ZFA set models under various conditions, extending known results from ZFC to alternative set theories.

Findings

In ZFA, the 'undirected' graph with loops is a random loopy graph.

Keeping multiple edges results in a non-aleph_0-categorical graph with many 1-types.

Disregarding single edges yields graphs with components isomorphic to any finite connected graph with loops.

Abstract

It is known that, if we take a countable model of Zermelo--Fraenkel set theory ZFC and "undirect" the membership relation (that is, make a graph by joining $x$ to $y$ if either $x\in y$ or $y\in x$), we obtain the Erd\H{o}s--R\'enyi 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 (if $x\in x$ for some $x$) or multiple edges (if $x\in y$ and $y\in x$ for some $x,y$). We 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. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.