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David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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British Journal for the Philosophy of Science 4 (14):107-129 (1953)
IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist) criticisms of classical logic, which Hilbert's theory was intended to meet, never even alluded to inconsistencies (in classical arithmetic), and since the investigations of Hilbert's school have always established much more than mere consistency, it is natural to formulate another general problem in the foundations of mathematics: to translate statements of theorems and proofs in the branch considered into those of some preferred system, where the translation must satisfy certain appropriate conditions (interpretation). The problem is relative to the choice of preferred system, as is Hilbert's consistency problem since he required the consistency to be established by particular methods (finitist ones). A finitist interpretation of classical number theory, which has been published in full detail elsewhere, is here described by means of typical examples. Partial results on analysis (theory of arbitrary functions whose arguments and values are the non-negative integers) are here presented for the first time. One of these results is restricted to functions whose values are bounded; its interest derives from the fact that real numbers may be represented by such functions. It is hoped that diverse general observations and comments, which would bore the specialist, may be of help to the general reader. The specialist may find some points of interest in the last two sections of the main text and in the notes following it
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Toshiyasu Arai (1998). Some Results on Cut-Elimination, Provable Well-Orderings, Induction and Reflection. Annals of Pure and Applied Logic 95 (1-3):93-184.
M. Hofmann, J. Van Oosten & T. Streicher (2006). Well-Foundedness in Realizability. Archive for Mathematical Logic 45 (7):795-805.
Harvey M. Friedman & Andre Scedrov (1986). Intuitionistically Provable Recursive Well-Orderings. Annals of Pure and Applied Logic 30 (2):165-171.
Yvon Gauthier (1971). Logique mathématique et philosophie des mathématiques. Dialogue 10 (2):243-275.
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