Gödel on concepts

Abstract : It is a well-known Gödelian thesis, since Hao Wang book of 1996, that a pure theory of concepts is the central part of logic (considered as the theoretical discipline of all Grundlagenforschungen) and that it cannot be reduced to set theory. Mathematics, according to Gödel, has to do with objects, but logic (in the Leibnizian, Fregean sense, namely as the unifying framework of all knowledge) is more than mathematics; in fact, the notion of concept is treated by logic in its most general sense. For these reasons, as Gödel explicitly states at the end of his 1944 paper on Russell, even if axiomatic set theory and the simple theory of types have been successful, at least to the extent that they permit the derivation of modern mathematics, many indications show that their primitive concepts require further elucidation. Gödel's assertion that a solution for the Continuum Hypothesis should be found by way of a pure theory of concepts, although unusual, reveals the importance that he assigned to the notion of concept. This article is an attempt to present Gödel's discussion on concepts, from 1944 to the late 1970s, in particular relation to the thought of Frege and Russell. The discussion takes its point of departure from Gödel's claim in notes on Bernays's review of ‘Russell's mathematical logic’. It then retraces the historical background of the notion of intension which both Russell and Gödel use, and offers some grounds for claiming that Gödel consistently considered logic as a free-type theory of concepts, called intensions, considered as the denotations of predicate-names.
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Gabriella Crocco. Gödel on concepts. History and Philosophy of Logic, Taylor & Francis, 2006, 27, pp.171-191. ⟨halshs-00094607⟩

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