A Normalizing Intuitionistic Set Theory with Inaccessible Sets

We propose a set theory strong enough to interpret powerful type theories underlying proof assistants such as LEGO and also possibly Coq, which at the same time enables program extraction from its constructive proofs. For this purpose, we axiomatize an impredicative constructive version of Zermelo-Fraenkel set theory IZF with Replacement and $\omega$-many inaccessibles, which we call \izfio. Our axiomatization utilizes set terms, an inductive definition of inaccessible sets and the mutually recursive nature of equality and membership relations. It allows us to define a weakly-normalizing typed lambda calculus corresponding to proofs in \izfio according to the Curry-Howard isomorphism principle. We use realizability to prove the normalization theorem, which provides a basis for program extraction capability.Comment: To be published in Logical Methods in Computer Scienc

De Jongh's Theorem for Intuitionistic Zermelo-Fraenkel Set Theory

We prove that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional logic IPC. More generally, we show that IZF has the de Jongh property with respect to every intermediate logic that is complete with respect to a class of finite trees. The same results follow for CZF.Comment: 12 page

Computer theorem proving in math

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003)

The limits of computation

This article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algo- rithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions