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 some peculiar aspects of the constructive theory of point-free spaces

This paper presents several independence results concerning the topos-valid and the intuitionistic (generalized) predicative theories of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined

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

Relational Parametricity for Computational Effects

According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Many programs, however, exhibit computational effects, and are not accounted for by the standard theory of relational parametricity. In this paper, we develop a foundational framework for extending the notion of relational parametricity to programming languages with effects.Comment: 31 pages, appears in Logical Methods in Computer Scienc