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types

In mathematics, logic, and computer science, a type theory is a formal system in which every "term" has a "type". A "type" in type theory has a role similar to a "type" in a programming language: it dictates the operations that can be performed on a term and, for variables, the possible values it might be replaced with.
Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions.
Type theory is closely related to, and in some cases overlaps with, type systems, which are a programming language feature used to reduce bugs and facilitate certain compiler optimizations. Because type theory and type systems can overlap, some experts use the phrase "type system" to refer to a specific formal system and the phrase "type theory" to refer to the academic study of them.

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