Typetheoretic

Typetheoretic refers to type theory, a family of formal systems that classify every term with a type and develop logic and computation on these typed objects. Type theory originated as an alternative foundation for mathematics and as a framework for programming languages, emphasizing constructive content and ensuring terms have well-defined interpretations. Central ideas include a hierarchy of types and terms, judgments such as Γ ⊢ t : T, and the use of dependent types where types may depend on terms. The Curry–Howard correspondence establishes a deep connection between proofs and programs: a proof of a proposition corresponds to a program of a type.

Key variants range from the simply typed lambda calculus to more expressive systems with dependent types and universes. Notable milestones include System F (polymorphic typed lambda calculus), Martin-Löf type theory (a constructive dependent type theory), and the Calculus of Constructions, which underpins the Coq and Lean families of proof assistants through the Calculus of Inductive Constructions. Type theories support inductive types, universes, and reflection, enabling formalization of mathematics and verified software.

Applications include formal verification of algorithms and hardware, formalized mathematics (proofs in Coq, Agda, Lean), and language design for statically checked, expressive programs. Type theory contrasts with set-theoretic foundations by focusing on computational content and constructive validity, though many type theories are classical in their logic or incorporate classical axioms. Critics note complexity and the generous expressiveness required to maintain consistency.