Adam-Day, B orcid.org/0000-0002-7891-916X and Cameron, PJ (2021) Undirecting membership in models of Anti-Foundation. Aequationes Mathematicae, 95 (2). pp. 393-400. ISSN 0001-9054
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∈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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2020. This is an open access article under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) (https://creativecommons.org/licenses/by/4.0/ ) |
Keywords: | Anti-Foundation Axiom; Erdős–Rényi random graph; Zermelo–Fraenkel axioms |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 09 Nov 2020 14:32 |
Last Modified: | 25 Jun 2023 22:29 |
Status: | Published |
Publisher: | Springer |
Identification Number: | 10.1007/s00010-020-00763-w |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:167743 |
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