Aczel's anti-foundation axiom (original) (raw)
In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel, as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to exactly one set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom is necessarily a non-well-founded set theory.
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| dbo:abstract | In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel, as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to exactly one set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom is necessarily a non-well-founded set theory. (en) L’axiome d’anti-fondation (en anglais, anti-foundation axiom ou AFA) est un axiome alternatif à l'axiome de fondation de la théorie des ensembles qui permet des chaînes infinies descendantes pour la relation d'appartenance sur les ensembles. Il permet par exemple à un ensemble d'appartenir à lui-même ou à deux ensembles distincts d'appartenir l'un à l'autre.Proposé par Marco Forti et Furio Honsell en 1983, il a été popularisé par l'ouvrage Non-Well-Founded Sets de Peter Aczel, publié en 1988. C'est un axiome qui propose une extension de l'ontologie ensembliste. En effet dans un univers de la théorie ZF (sans axiome de fondation) il est toujours possible de définir une partie de celui-ci, l'univers de von Neumann, qui satisfait tous les axiomes de ZF et l'axiome de fondation, ce sont les ensembles bien fondés. L'axiome d'anti-fondation a pour conséquence que l'univers de von Neumann n'est pas l'univers tout entier : il existe des ensembles non-bien fondés (appelés parfois hyper-ensembles). Cette vision avait été anticipée par le mathématicien Dmitry Mirimanoff. (fr) |
| dbo:wikiPageExternalLink | http://www.cs.bilkent.edu.tr/~akman/jour-papers/jiis/jiis.pdf http://www.goertzel.org/books/logic/contents.html http://www.goertzel.org/books/logic/chapter_seven.htm https://archive.org/details/nonwellfoundedse0000acze%7Caccess-date= |
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| dbp:authorlink | Peter Aczel (en) |
| dbp:first | Peter (en) |
| dbp:last | Aczel (en) |
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| dbp:year | 1988 (xsd:integer) |
| dct:subject | dbc:Directed_graphs dbc:Axioms_of_set_theory |
| gold:hypernym | dbr:Axiom |
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| rdfs:comment | In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel, as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to exactly one set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom is necessarily a non-well-founded set theory. (en) L’axiome d’anti-fondation (en anglais, anti-foundation axiom ou AFA) est un axiome alternatif à l'axiome de fondation de la théorie des ensembles qui permet des chaînes infinies descendantes pour la relation d'appartenance sur les ensembles. Il permet par exemple à un ensemble d'appartenir à lui-même ou à deux ensembles distincts d'appartenir l'un à l'autre.Proposé par Marco Forti et Furio Honsell en 1983, il a été popularisé par l'ouvrage Non-Well-Founded Sets de Peter Aczel, publié en 1988. (fr) |
| rdfs:label | Aczel's anti-foundation axiom (en) Axiome d'anti-fondation (fr) |
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