Lesson plan

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 .

Back to article

Article summary

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 .

Key claims

No claims recorded.

Discussion questions

  1. Summarise the article in your own words. Which single claim would you verify first, and why?
  2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
  3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

Bibliography

No sources recorded.

Key claims and bibliography are generated from the same tracked claims and sources as the article page.