nLab reflexive set

Definition

In unsorted set theory

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

$$X \in X .$$

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

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

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:

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

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

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

Examples

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 $1$ 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 $1$, and elements are represented by morphisms with domain $1$. Thus, we define the predicates $\mathrm{set}(f) \coloneqq \mathrm{codom}(f) = 1$ and $\mathrm{element}(f) \coloneqq \mathrm{dom}(f) = 1$. The codomain and domain of general functions are then defined as usual to be sets, $\mathrm{set}(\mathrm{codom}(f))$ and $\mathrm{set}(\mathrm{dom}(f))$. We define the membership relation $a \in b$ as requiring the morphism $a$ to be an element, the morphism $b$ to be a set, and the codomain of $a$ to be $b$:

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

The terminal object $1$ is then a Quine atom with respect to $\in$.

See also

• unsorted set theory
• fully formal ETCS