„Quine-Atom“ – Versionsunterschied
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== Quine atoms<!--'Quine atom' and 'Reflexive set' redirect here--> == | == Quine atoms<!--'Quine atom' and 'Reflexive set' redirect here--> == | ||
https://en.wikipedia.org/wiki/Urelement#Quine_atoms | |||
An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set. '''Quine atoms'''<!--boldface per WP:R#PLA--> (named after [[Willard Van Orman Quine]]) are sets that only contain themselves, that is, sets that satisfy the formula ''x'' = {''x''}.<ref name="Forster2003">{{cite book|author=Thomas Forster|title=Logic, Induction and Sets|url=https://books.google.com/books?id=mVeTuaRwWssC&pg=PA199|year=2003|publisher=Cambridge University Press|isbn=978-0-521-53361-4|page=199}}</ref> | An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set. '''Quine atoms'''<!--boldface per WP:R#PLA--> (named after [[Willard Van Orman Quine]]) are sets that only contain themselves, that is, sets that satisfy the formula ''x'' = {''x''}.<ref name="Forster2003">{{cite book|author=Thomas Forster|title=Logic, Induction and Sets|url=https://books.google.com/books?id=mVeTuaRwWssC&pg=PA199|year=2003|publisher=Cambridge University Press|isbn=978-0-521-53361-4|page=199}}</ref> | ||
Version vom 8. Juli 2025, 12:35 Uhr
Quine atoms
https://en.wikipedia.org/wiki/Urelement#Quine_atoms
An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set. Quine atoms (named after Willard Van Orman Quine) are sets that only contain themselves, that is, sets that satisfy the formula x = {x}.[1]
Quine atoms cannot exist in systems of set theory that include the axiom of regularity, but they can exist in non-well-founded set theory. ZF set theory with the axiom of regularity removed cannot prove that any non-well-founded sets exist (unless it is inconsistent, in which case it will prove any arbitrary statement), but it is compatible with the existence of Quine atoms. Aczel's anti-foundation axiom implies that there is a unique Quine atom. Other non-well-founded theories may admit many distinct Quine atoms; at the opposite end of the spectrum lies Boffa's axiom of superuniversality, which implies that the distinct Quine atoms form a proper class.[2]
Quine atoms also appear in Quine's New Foundations, which allows more than one such set to exist.[3]
Quine atoms are the only sets called reflexive sets by Peter Aczel,[2] although other authors, e.g. Jon Barwise and Lawrence Moss, use the latter term to denote the larger class of sets with the property x ∈ x.[4]
- ↑ Thomas Forster: Logic, Induction and Sets. Cambridge University Press, 2003, ISBN 978-0-521-53361-4, S. 199 (google.com).
- ↑ a b Vorlage:citation.
- ↑ Vorlage:citation.
- ↑ Vorlage:citation.