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We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is(More)
We explain in detail why the notion of list-arithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by Andrè Joyal to give a categorical proof of Gödel's incompleteness results. We motivate this definition for three reasons: first, Joyal's arithmetic universes are list-arithmetic(More)
We reconsider Rauszer's bi-intuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of polarized bi-intuitionistic logic (PBL) consists of two fragments,(More)
In this paper we analyze an extension of Martin-Löf's intensional set theory by means of a set contructor P such that the elements of P(S) are the subsets of the set S. Since it seems natural to require some kind of extensionality on the equality among subsets, it turns out that such an extension cannot be constructive. In fact we will prove that this(More)
There are several kinds of linear typed calculus in the literature, some with their associated notion of categorical model. Our aim in this paper is to systematise the relationship between three of these linear typed calculi and their models. We point out that mere sound-ness and completeness of a linear typed calculus with respect to a class of categorical(More)
We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hy-perdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a(More)