“Map Theory et Antifondation”

Abstract

Map Theory is a powerful extension of type-free lamba-calculus (with only a few term constants added). Due to Klaus Grue, it was designed to be a common foundation for Computer Sciences and for Mathematics. All the primitive notions of first-order logic and set theory, including truth values, connectives and quantifiers, set-membership and set-equality, are interpreted as terms. All the usual set-theoretic constructs, including inductive data-types, get computational interpretations.Now, Grue's version of Map Theory is founded, in the sense that it only considers mathematical sets or classes which are well-founded with respect to the membership relation. Indeed, it was shown to be at least as powerful as ZFC+FA, where FA is the usual well-foundation axiom of set theory.In this thesis, we show that it is possible to design an alternative version which takes non-well-founded sets into account, and allows for co-inductive reasoning over them. This new version opens the way to a direct representation of co-inductive data-types and of circular processes and phenomena in Map Theory. 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 particular, this interpretation implies the embedding of the first order reasoning on formulas into equational reasoning on terms which translate these formulas. In the second part, we show the consistency of MTA inside the framework of the k-continuous semantics for k > σ, where σ is any strongly inaccessible cardinal. The proof uses the k-premodels of Berline-Grue (k-cpos satisfying few simple additional properties). The main issue in modelling MTA is the construction, inside any k-premodel M, of an adequate k-open subset Φ of M which, once enriched with adequate equality and membership relations, will be a model of ZFC+AFA.