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dc.contributor.authorAdam-Day, Bea
dc.contributor.authorCameron, Peter J.
dc.identifier.citationAdam-Day , B & Cameron , P J 2021 , ' Undirecting membership in models of Anti-Foundation ' , Aequationes Mathematicae , vol. 95 , no. 2 , pp. 393-400 .
dc.identifier.otherPURE: 271141935
dc.identifier.otherPURE UUID: 20eb0c5c-2ef2-4501-860f-79f52e9f48d2
dc.identifier.otherORCID: /0000-0003-3130-9505/work/83889571
dc.identifier.otherScopus: 85096210210
dc.identifier.otherWOS: 000590486200001
dc.description.abstractIt 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 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 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.
dc.relation.ispartofAequationes Mathematicaeen
dc.rightsCopyright © The Author(s) 2020. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
dc.subjectRandom graphen
dc.subjectAnti-Foundation Axiomen
dc.subjectErdős–Rényi random graphen
dc.subjectZermelo–Fraenkel axiomsen
dc.subjectQA Mathematicsen
dc.titleUndirecting membership in models of Anti-Foundationen
dc.typeJournal articleen
dc.description.versionPublisher PDFen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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