nLab inhabited set

Idea

A set or type is inhabited if it contains an element or term.

Definition

In set theory

In set theory, an inhabited set is a set that contains an element, i.e. a set $X$ such that $\exists x, x\in X$ is true.

At least assuming classical logic, this is the same thing as a set that is not empty. Usually inhabited sets are simply called ‘non-empty’, but the positive word ‘inhabited’ reminds us that inhabitation is the simpler notion, which emptiness is defined as the negation of.

The term ‘inhabited’ comes from constructive mathematics. In constructive mathematics (such as the internal logic of some topos or generally in type theory), a set/type that is not empty is not already necessarily inhabited. This is because double negation is nontrivial in intuitionistic logic. All the same, many constructive mathematicians use the old word ‘non-empty’ with the understanding that it really means inhabited, and write $A\neq \emptyset$ to mean that $A$ is inhabited. The latter we can interpret literally if we regard $\neq$ as a reference to an inequality relation other than the denial inequality, such as the inequality defined for subsets by $A \neq B \iff \exists x ((x\in A \wedge x\notin B)\vee(x\in B\wedge x\notin A))$. If we prefer to reserve $\neq$ for the denial inequality, then we can write $\#$ for this stronger inequality of sets (although it is not an apartness relation), and hence $A\#\emptyset$ to mean that $A$ is inhabited.

In type theory

In type theory there are two possible notions of inhabited type: a type $X$ whose propositional truncation $\Vert X \Vert$ has an element (or term), or a type $X$ that itself has an element (or term). The former is what corresponds to the above notion of inhabitedness in set theory, since $\Vert X \Vert$ is the propositions as types interpretation of $\exists x:X$. The latter is more like the notion of a pointed set.

The assertion $\forall X, (\Vert X \Vert \to X)$ is a mildly nonconstructive logical principle called the propositional axiom of choice. It follows from excluded middle, but in the internal logics of some toposes, it can fail, so that these two notions of “inhabited type” really are different.

Inhabited objects

An inhabited set is the special case of an internally inhabited object in the topos Set. The two notions of inhabited type correspond to internally and externally inhabited objects (which are, respectively, those objects $X$ where $X\to 1$ is an epimorphism, and those which admit a global element $1\to X$).

Related notions

There is a distinction between ‘inhabited’ and ‘occupied?’ spaces in Abstract Stone Duality (which probably corresponds to something about locales, should explain that here).

Related concepts

• hProp

• effective epimorphism

• inhabited object

• bracket type, image

• occupied space?