intuitionistic type theory

Intuitionistic type theory is a formal system within intuitionism, a school of mathematical thought emphasizing mental constructions. Introduced by L.E.J. Brouwer, intuitionism challenges classical logic's principle of the excluded third, particularly with infinite sets, arguing that mathematical statements must be constructively proven.1

Intuitionistic type theories, like (\mathcal{T}_0), are designed to align with this constructive approach. They include types for truth values ((\Omega)), natural numbers ((N)), and sets of entities of a given type ((\mathcal{P}(A))). Key terms include formulas for equality and membership, numerals, and comprehension terms for defining sets based on properties. This framework differs from classical type theory by omitting the axiom of double negation elimination. While powerful for constructive mathematics, classical arguments are often still needed for metamathematics.213