Can You Add Power-Sets to Martin-Löf's Intuitionistic Set Theory?

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...

Kripke Semantics for Martin-Löf's Extensional Type Theory

It is well-known that simple type theory is complete with respect to nonstandard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of inte...

A cartesian closed category in Martin-Löf's intuitionistic type theory

First, we briefly recall the main definitions of the theory of Information Bases and Translations. These mathematical structures are the basis to construct the cartesian closed category InfBas, which is equivalent to the category ScDom of Scott Domains. Then, we will show that all the definitions and the proof of all the properties that one needs in order to show that InfBas is indeed a cartesi...

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

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

We prove that every strictly positive endofunctor on the category of sets generated by Martin-LL of's extensional type theory has an initial algebra. This representation of inductively deened sets uses essentially the wellorderings introduced by Martin-LL of in \Constructive Mathematics and Computer Programming". 1 Background Martin-LL of 10] introduced a general set former for wellorderings in...