"How did the serpent of inconsistency enter Frege's paradise?"

Abstract

My project here is the appraisal of Michael Dummett's diagnosis that the inconsistency of Frege’s system in Grundgesetze is attributable to his neglect of the indefinite extensibility of fundamental mathematical concepts. I address a problem that obscures the usual intuitive characterisations of the notion of indefinite extensibility, and offer thereby what I believe to be the correct characterisation of the notion. En passant, some issues are reviewed about the "size" of indefinitely extensible concepts. And that will bring us into position to scrutinise the connections of the notion as characterised with paradox, and specifically the paradox that Russell found for Frege’s Basic Law V. It will be argued that Dummett’s diagnosis is apt for the Burali-Forti paradox, wrong for Cantor’s paradox, and correct for the paradox inherent in Law V only on the assumption that full classical imperative higher-order logic is indeed wholly logical.