„Quine-Atom“ – Versionsunterschied
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'Quine atom' and 'Reflexive set' redirect here | |||
== Quine atoms == | |||
https://en.wikipedia.org/wiki/Urelement#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''' | 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''}. | ||
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 [[Principle of explosion|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]]. | 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 [[Principle of explosion|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]]. | ||
Quine atoms also appear in Quine's [[New Foundations]], which allows more than one such set to exist. | Quine atoms also appear in Quine's [[New Foundations]], which allows more than one such set to exist. | ||
Quine atoms are the only sets called '''reflexive sets''' | Quine atoms are the only sets called '''reflexive sets''' by [[Peter Aczel]], 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''. | ||
Aktuelle Version vom 11. August 2026, 05:16 Uhr
'Quine atom' and 'Reflexive set' redirect here
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}.
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.
Quine atoms also appear in Quine's New Foundations, which allows more than one such set to exist.
Quine atoms are the only sets called reflexive sets by Peter Aczel, 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.