Foundations of Science 18 (4):655-680 (2013)

Peter Verdee
Université Catholique de Louvain
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
Keywords Adaptive logic  Foundations of mathematics  Comprehension  Pragmatic foundations  Set theory  Non-classical logic
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DOI 10.1007/s10699-012-9296-5
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References found in this work BETA

Saving Truth From Paradox.Hartry Field - 2008 - Oxford University Press.
A Universal Logic Approach to Adaptive Logics.Diderik Batens - 2007 - Logica Universalis 1 (1):221-242.

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Citations of this work BETA

Remarks on Naive Set Theory Based on Lp.Hitoshi Omori - 2015 - Review of Symbolic Logic 8 (2):279-295.
Carnapian Structuralism.Holger Andreas - 2014 - Erkenntnis 79 (S8):1373-1391.
Adaptive Fregean Set Theory.Diderik Batens - 2020 - Studia Logica 108 (5):903-939.

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