Type (mathematics)

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type.

Type theory is closely related to (and in some cases overlaps with) type systems, which are a programming language feature used to reduce bugs. Type theory was created to avoid paradoxes in a variety of formal logics and rewrite systems.

Two well-known type theories, that can serve as mathematical foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory.

Between 1902 and 1908 Bertrand Russell proposed various "theories of type" in response to his discovery that Gottlob Frege's version of naive set theory was afflicted with Russell's paradox. By 1908 Russell arrived at a "ramified" theory of types together with an "axiom of reducibility" both of which featured prominently in Whitehead and Russell's Principia Mathematica published between 1910 and 1913. They attempt to avoid Russell's paradox by first creating a hierarchy of types, then assigning each mathematical (and possibly other) entity to a type. Objects of a given type are built exclusively from objects of preceding types (those lower in the hierarchy), thus preventing loops. In the 1920s, Leon Chwistek and Frank P. Ramsey proposed a simpler theory, now known as the "theory of simple types" or "simple type theory", that collapsed the complicated hierarchy of the "ramified theory" and did not require the axiom of reducibility.

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