In
type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a found ...
, the empty type or absurd type, typically denoted
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
For any type
, the type
is defined as
. As the notation suggests, by the
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relati ...
, a term of type
is a false proposition, and a term of type
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,
is also
uninhabited
The list of uninhabited regions includes a number of places around the globe. The list changes year over year as human beings migrate into formerly uninhabited regions, or migrate out of formerly inhabited regions.
List
As a group, the list of ...
for any inhabited type
.
If a type system contains an empty type, the
bottom type
In type theory, a theory within mathematical logic, the bottom type of a type system is the type that is a subtype of all other types.
Where such a type exists, it is often represented with the up tack (⊥) symbol.
When the bottom type is empty, ...
must be uninhabited too, so no distinction is drawn between them and both are denoted
.
References
Type theory
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