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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higher-order primitive recursion and higher-order intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigma-types of the calculus enable us to achieve this. There are many similarities with the subset interpretation of Martin-Lof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...

. To any entailment relation [Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of some spaces associated to mathematical structures. 1 Introduction Most spaces associated to mathematical structures: spectrum of a ring, space of valuations of a field, space of bounded linear functionals, . . . can be represented as distributive lattices. The key to have a natural definition in these cases is to use the notion of entailment relation due to Dana Scott. This note explains the connection between entailment relations and distributive lattices. An entailment relation may be seen as a logical description of a distributive lattice. Furthermore, most operations on distributive lattices are simpler when formulated as operations on entailment relations. A special kind of distribu...

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

By a classifying topos for a first-order theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate `smallness condition', and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every first-order theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heyting-valued completeness theorem for infinitary first-order logic.

this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a suitable formal topology, and the "existence" of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topological proof [6, 8], but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements. 1 Construction of Minimal Invariant Subspace

In this paper we study the logic L !! , which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of L !! with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting valued models. The logic L !! is the strongest for which Heyting valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.

Abstract. Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.?1. Introduction. In 1963, Paul Cohen introduced the method of forcing to prove the independence of both the axiom of choice and the continuum hypothesis from Zermelo-Fraenkel set theory. It was not long before Saul Kripke noted a connection between forcing and his semantics for modal and

This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions of some of the less standard terms used in the paper, but the reader is expected to be familiar with the basic ideas of category theory. A rough machine generated index begins on page 21. I would have liked to explain the main ideas of all the papers referred to herein, but I am not familiar enough with some of them to do that. It seemed more useful to be inclusive, even if many papers were mentioned without comment. One consequence of this is that the discussions in this document often go into more detail about the papers published in North America than about those published elsewhere. The DVI file for this article is available by anonymous FTP from ftp.cwru.edu in the directory

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