Quine-Atom
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.