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Map Theory et Antifondation

Abstract : 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.
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Contributor : Thierry Vallée <>
Submitted on : Friday, April 5, 2002 - 2:17:41 PM
Last modification on : Friday, March 27, 2020 - 3:15:49 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 5:15:24 PM


  • HAL Id : tel-00001298, version 1



Thierry Vallée. Map Theory et Antifondation. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2001. Français. ⟨tel-00001298⟩



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