nLab reflexive set



In unsorted set theory

In unsorted set theory, a reflexive set is a set that belongs to itself:

XX. X \in X .

Equivalently, XX is reflexive iff it equals its successor X{X}X \cup \{X\}. 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:

X={X}. X = \{X\} .

In two-sorted set theory

In two-sorted set theory with element reflection, a reflexive set is a set whose element reflection belongs to itself:

asElem(X)X\mathrm{asElem}(X) \in X

Similarly, in a two-sorted set theory with set reflection, a reflexive element is an element which belongs to its set reflection:

XasSet(X)X \in \mathrm{asSet}(X)


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.

Fully formal ETCS

Take any single-sorted definition of a well-pointed topos \mathcal{E}, such as fully formal ETCS, which by definition has a morphism 11 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 11, and elements are represented by morphisms with domain 11. Thus, we define the predicates set(f)codom(f)=1\mathrm{set}(f) \coloneqq \mathrm{codom}(f) = 1 and element(f)dom(f)=1\mathrm{element}(f) \coloneqq \mathrm{dom}(f) = 1. The codomain and domain of general functions are then defined as usual to be sets, set(codom(f))\mathrm{set}(\mathrm{codom}(f)) and set(dom(f))\mathrm{set}(\mathrm{dom}(f)). We define the membership relation aba \in b as requiring the morphism aa to be an element, the morphism bb to be a set, and the codomain of aa to be bb:

abelement(a)set(b)codom(a)=ba \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = b

The terminal object 11 is then a Quine atom with respect to \in.

See also

Last revised on October 24, 2022 at 22:57:32. See the history of this page for a list of all contributions to it.