A synthetic axiomatization of Map Theory

Includes TOC détaillée, index et appendicesInternational audienceThis paper presents a subtantially simplified axiomatization of Map Theory and proves the consistency of this axiomatization in ZFC under the assumption that there exists an inaccessible ordinal. Map Theory axiomatizes lambda calculus plus Hilbert's epsilon operator. All theorems of ZFC set theory including the axiom of foundation are provable in Map Theory, and if one omits Hilbert's epsilon operator from Map Theory then one is left with a computer programming language. Map Theory fulfills Church's original aim of introducing lambda calculus. Map Theory is suited for reasoning about classical mathematics as well ascomputer programs. Furthermore, Map Theory is suited for eliminating thebarrier between classical mathematics and computer science rather than just supporting the two fields side by side. Map Theory axiomatizes a universe of "maps", some of which are "wellfounded". The class of wellfounded maps in Map Theory corresponds to the universe of sets in ZFC. The first version MT0 of Map Theory had axioms which populated the class of wellfounded maps, much like the power set axiom et.al. populates the universe of ZFC. The new axiomatization MT of Map Theory is "synthetic" in the sense that the class of wellfounded maps is defined inside MapTheory rather than being introduced through axioms. In the paper we define the notion of kappa- and kappasigma-expansions and prove that if sigma is the smallest strongly inaccessible cardinal then canonical kappasigma expansions are models of MT (which proves the consistency). Furthermore, in the appendix, we prove that canonical omega-expansions are fully abstract models of the computational part of Map Theory

The implementation of Logiweb

This paper describes the implementation of the ‘Logiweb ’ system with emphasis on measures taken to support classical reasoning about programs. Logiweb is a system for authoring, storing, distributing, indexing, checking, and rendering of ‘Logiweb pages’. Logiweb pages may contain mathematical definitions, conjectures, lemmas, proofs, disproofs, theories, journal papers, computer programs, and proof checkers. Reading Logiweb pages merely requires access to the World Wide Web. Two example pages are available o

Map theory et antifondation

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Map Theory et Antifondation

Map Theory is a powerful extension of type-free lamba-calculus. Due to Klaus Grue, it was designed to be a common foundation for Computer Sciences and for Mathematics. In particular Map Theory interprets predicate calculus and ZFC+FA , where ZFC is the theory of Zermelo-Fraenkel, and FA is the usual well-foundation axiom. All the primitive notions of first-order logic and set theory, including truth values, connectives and quantifiers, set-membership and set-equality, get a canonical interpretation as terms of the lambda-calculus with only a few term constants added. Moreover, Map Theory allows to represent inductive data type and gives a computational interpretation to all the usual set-theoretic constructs. K. Grue's version of Map Theory only considers mathematical sets or classes which are well-founded with respect to the membership relation. In this thesis we show that it is possible to design a version of Map Theory which takes all non-well-founded sets into account, and allows for co-inductive reasoning over them. This new system opens the way to a direct representation of co-inductive data-types and of circular processes and phenomena. In the first part of the thesis we present the axiomatization of this new system, called MTA, and we show that it is powerful enough to interpret ZFC+AFA, where AFA is the Aczel-Forti-Honsell Antifoundation axiom. In the second part, we show the relative consistency of MTA with respect to ZFC+SI, where SI is the axiom which forces the existence of a strongly inaccessible cardinal.Map Theory est une extension équationnelle du lambda-calcul non-typé conçue par Klaus Grue pour être une fondation commune de l'informatique et des mathématiques. Elle permet en particulier une interprétation complète du calcul des prédicats et de ZFC+FA, où ZFC est la théorie de Zermelo-Fraenkel, et FA est l'axiome de bonne fondation usuel. Toutes les notions primitives de la logique du premier ordre et de la théorie des ensembles, valeurs de vérité, connecteurs, quantificateurs, appartenance et égalité, y sont traduites par des termes du lambda-calcul enrichi de quelques constantes. De plus, Map Theory permet de représenter les types de données inductifs et de donner un sens calculatoire immédiat à tous les constructeurs ensemblistes usuels. La version initiale de Map Theory par K. Grue ne considère cependant que les ensembles (ou classes) bien-fondée relativement à la relation d'appartenance. Dans le cadre du renouveau d'intérêt pour l'antifondation induit par les developpements récents de l'informatique théorique, nous montrons dans notre thèse qu'il est possible d'élaborer une version antifondée de Map Theory qui prenne en compte l'existence des objets non-bien-fondés, et qui permette de raisonner sur ces objets par co-induction. Ce nouveau système ouvre la possibilité d'une représentation directe des types de données co-inductifs, et de la modèlisation des phènoménes et processus circulaires. Dans une première partie, nous présenterons l'axiomatisation MTA de ce nouveau système, et nous montrerons que ZFC+AFA, où AFA est l'axiome d'Antifondation de Aczel-Forti-Honsell, y est interprétable syntaxiquement. Dans la deuxième partie, nous montrerons la consistance de MTA relativement à ZFC+SI, où SI est l'axiome exprimant l'existence d'un cardinal fortement inaccessible