CZF does not have the Existence Property

Constructive theories usually have interesting metamathematical properties where explicit witnesses can be extracted from proofs of existential sentences. For relational theories, probably the most natural of these is the existence property, EP, sometimes referred to as the set existence property. This states that whenever (\exists x)\phi(x) is provable, there is a formula \chi(x) such that (\exists ! x)\phi(x) \wedge \chi(x) is provable. It has been known since the 80's that EP holds for some intuitionistic set theories and yet fails for IZF. Despite this, it has remained open until now whether EP holds for the most well known constructive set theory, CZF. In this paper we show that EP fails for CZF

On Tarski's fixed point theorem

A concept of abstract inductive definition on a complete lattice is formulated and studied. As an application, a constructive and predicative version of Tarski's fixed point theorem is obtained.Comment: Proc. Amer. Math. Soc., to appea

Constructive Mathematics in Theory and Programming Practice

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop’s constructive mathematics(BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focuses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof’s theory of types as a formal system for BISH