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De quoi les "théorèmes de limitation des formalismes" : théorèmes de Gödel de 1931 et apparentés, sont-ils la limitation?

Abstract : We want to define the limitations content revealed by the theorems of formalisms limitation (Godel's theorems of 1931, Church's theorem of 1936 and Turing's theorem of 1936-1937). In order to answer this question, we have accepted as main theme Hilbert' s program (in the broad sense) : on the one hand, the answer that Hilbert hoped to give to foundations problem, and on the other hand, the justification he hoped to give to the lack of insoluble mathematical problems. This first lead us to propose a precise interpretation of the two aspects of this program. We have then analyzed the various proposals which have been given in answer this program, including in particular Michael Detlefsen'one, taking into account arithmetical indecidability results obtained in the 1970's. In this aim we have made a detailed analysis of Church-Turing's thesis. We have also discussed the different positions which have been held within the framework induced by Lucas-Penrose's argument. We have then discussed Post, Myhill and Ladrière's successively answers given to the general question asked. On the basis on this whole analysis, our own answer is that these theorems show a kind of relativity in relation with the use of formalization itself, which must be rooted in a confined part of the empirical practice of informal mathematics.
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Submitted on : Thursday, February 27, 2020 - 5:58:10 PM
Last modification on : Tuesday, September 22, 2020 - 3:46:08 AM


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Patrice Pissavin. De quoi les "théorèmes de limitation des formalismes" : théorèmes de Gödel de 1931 et apparentés, sont-ils la limitation?. Philosophie. Université Panthéon-Sorbonne - Paris I, 2019. Français. ⟨NNT : 2019PA01H212⟩. ⟨tel-02493455⟩



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