Undirecting membership in models of Anti-Foundation

Bea Adam-Day, Peter J. Cameron

Research output: Contribution to journalArticlepeer-review


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∈y or y∈x), we obtain the Erdős–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 (if x∈x for some x) or multiple edges (if x∈y and y∈x for some distinct xy). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is ℵ0-categorical and homogeneous), but if we keep multiple edges, the resulting graph is not ℵ0-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.
Original languageEnglish
Pages (from-to)393-400
Number of pages8
JournalAequationes Mathematicae
Issue number2
Early online date18 Nov 2020
Publication statusPublished - Apr 2021


  • Random graph
  • Anti-Foundation Axiom
  • Erdős–Rényi random graph
  • Zermelo–Fraenkel axioms


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