Grim's arguments against omniscience and indefinite extensibility

Abstract
Patrick Grim has put forward a set theoretical argument purporting to prove that omniscience is an inconsistent concept and a model theoretical argument for the claim that we cannot even consistently define omniscience. The former relies on the fact that the class of all truths seems to be an inconsistent multiplicity (or a proper class, a class that is not a set); the latter is based on the difficulty of quantifying over classes that are not sets. We first address the set theoretical argument and make explicit some ways in which it depends on mathematical Platonism. Then we sketch a non Platonistic account of inconsistent multiplicities, based on the notion of indefinite extensibility, and show how Grim’s set theoretical argument could fail to be conclusive in such a context. Finally, we confront Grim’s model theoretical argument suggesting a way to define a being as omniscient without quantifying over any inconsistent multiplicity.
Keywords Cantor’s theorem  Indefinite extensibility  Omniscience  Platonism  Truth  Universe of discourse
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DOI 10.1007/s11153-011-9301-x
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Matemáticas y Platonismo(S).J. Ferreiros - 1999 - Gaceta de la Real Sociedad Matemática Española 2 (446):473.
The Incomplete Universe.Patrick Grim - 1991 - Cambridge: Mass.: Mit Press.
The Problem of Absolute Universality.Charles Parsons - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 203--19.
Something About Everything: Universal Quantification in the Universal Sense of Universal Quantification.Shaughan Lavine - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 98--148.

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