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Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics

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Verdée, Peter [UCL]

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations.

Document type Article de périodique (Journal article)
Access type Accès restreint
Publication date 2012
Language Anglais
Journal information "Foundations of Science" - Vol. 18, no. 4, p. 655-680 (2012)
Peer reviewed yes
Publisher Springer Netherlands (Dordrecht)
issn 1233-1821
e-issn 1572-8471
Publication status Publié
Affiliation UCL - SSH/ISP - Institut supérieur de philosophie
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Bibliographic reference Verdée, Peter. Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics. In: Foundations of Science, Vol. 18, no. 4, p. 655-680 (2012)
Permanent URL http://hdl.handle.net/2078.1/164430