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 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 \(x\in x\) for some x) or multiple edges (if \(x\in y\) and \(y\in x\) for some distinct 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.

众所周知的是，如果我们采取策梅罗-弗兰克尔集合论ZFC的可数模型和“undirect”隶属关系（即，使通过加入的图形X到Y ^如果任一\（在Y X \ \）或\（ y \ in x \）），我们得到Erdős–Rényi随机图。证明这一点的关键公理是基金会公理。因此很自然地想知道如果我们删除此公理或将其替换为替代方案（例如Aczel的反基金会公理）会发生什么。结果图可能不简单。它可以具有环（如果\（X \ x中\）对于某些X）或多个边缘（如果\（X \在Y \）和\（Y \ x中\）为一些不同的X， y）。我们表明，在ZFA中，如果我们保留循环并忽略多个边，则会获得“随机循环图”（它是\（\ aleph _0 \）-分类且同质的），但是如果我们保留多个边，则生成的图不是\（\ aleph _0 \）-分类的，但是具有无限多个1型。此外，如果我们仅保留循环和双边而丢弃单边，则生成的图包含许多同构的连通分量，这些连通分量同任何给定的带有环的有限连通图同构。