12 Facts About 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.

Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic.

For example, when a type theory has a rule that defines the type "bool", it defines the function "if".

Openendedness of Martin-Lof type theory is particularly manifest in the introduction of so-called universes.

Complexities of equality in type theory make it an active research area, see homotopy type theory.

Type theory theories differ from this foundation in a number of ways.

Type theory is naturally associated with the decision problem of type inhabitation.

In 2016 cubical type theory was proposed, which is a homotopy type theory with normalization.

Type theory is often cited as an implementation of the Brouwer–Heyting–Kolmogorov interpretation.

Martin-Lof specifically developed intuitionistic type theory to encode all mathematics to serve as a new foundation for mathematics.

Mathematicians working in category Type theory already had difficulty working with the widely accepted foundation of Zermelo–Fraenkel set Type theory.

Much of the current research into type theory is driven by proof checkers, interactive proof assistants, and automated theorem provers.