Representing Inductively Defined Sets by Wellorderings in Martin-Löf's Type Theory

A Set Constructor for Inductive Sets in Martin-Löf's Type Theory

A Coherence Theorem for Martin-Löf's Type Theory

In type theory a proposition is represented by a type, the type of its proofs. As a consequence, the equality relation on a certain type is represented by a binary family of types. Equality on a type may be conventional or inductive. Conventional equality means that one particular equivalence relation is singled out as the equality, while inductive equality – which we also call identity – is in...

Set theoretical proofs as type theoretical programs

We show, that all Π 0 2-sentences, provable in the set theory KP I + U can be proved in Martin-Löf's Type Theory with W-type and one universe. Therefore set theoretical proofs can be considered as programs in type theory. The method used is a formalisation of proof theoretical methods in type theory. The result will be a high level type theory program using the full strength of Martin-Löf's Typ...

Errata to "Martin-Löf's Type Theory as an Open-Ended Framework"

This paper treats Martin-Löf’s type theory as an open-ended framework composed of (i) flexibly extensible languages into which various forms of objects and types can be incorporated, (ii) their uniform, effectively given semantics, and (iii) persistently valid inference rules. The class of expression systems is introduced here to define an openended body of languages underlying the theory. Each...

A Finite Axiomatization of Inductive-Recursive Definitions

Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Löf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order univers...