In unsorted set theory, a reflexive set is a set that belongs to itself:
Equivalently, is reflexive iff it equals its successor . Compare transitive sets.
If the axiom of foundation holds in a set theory, then there are no reflexive sets. In non-well-founded set theory, however, there may be many reflexive sets.
A Quine atom is a minimally reflexive set:
In two-sorted set theory with element reflection, a reflexive set is a set whose element reflection belongs to itself:
Similarly, in a two-sorted set theory with set reflection, a reflexive element is an element which belongs to its set reflection:
In Peter Aczel's ill-founded set theory, there is a unique Quine atom. On the other hand, by exempting Quine atoms (and only Quine atoms) from the axiom of foundation, one obtains a theory of pure sets equivalent to well-founded material sets with urelements.
Take any single-sorted definition of a well-pointed topos , such as fully formal ETCS, which by definition has a morphism representing the terminal object, the identity morphism of the terminal object, and the single global element of the terminal object. Sets are represented by morphisms with codomain , and elements are represented by morphisms with domain . Thus, we define the predicates and . The codomain and domain of general functions are then defined as usual to be sets, and . We define the membership relation as requiring the morphism to be an element, the morphism to be a set, and the codomain of to be :
The terminal object is then a Quine atom with respect to .
Last revised on October 24, 2022 at 22:57:32. See the history of this page for a list of all contributions to it.