Empty type

In type theory, an empty type or absurd type, typically denoted $$\mathbb 0$$ is a type with no terms. Such a type may be defined as the nullary coproduct (i.e. disjoint sum of no types). It may also be defined as the polymorphic type $$\forall t.t$$

For any type $$P$$, the type $$\neg P$$ is defined as $$P\to \mathbb 0$$. As the notation suggests, by the Curry–Howard correspondence, a term of type $$\mathbb 0$$ is a false proposition, and a term of type $$\neg P$$ is a disproof of proposition P.

A type theory need not contain an empty type. Where it exists, an empty type is not generally unique. For instance, $$T \to \mathbb 0$$ is also uninhabited for any inhabited type $$T$$.

If a type system contains an empty type, the bottom type must be uninhabited too, so no distinction is drawn between them and both are denoted $$\bot$$.