nLab
reflexive set

Reflexive sets

In material 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.

By the axiom of foundation, there are no reflexive sets. In ill-founded set? theory, however, there may be many reflexive sets.

A Quine atom is a minimally reflexive set:

X={X}. X = \{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.

Last revised on January 7, 2018 at 20:54:19. See the history of this page for a list of all contributions to it.