Chapter

Problems

David Bostock

in Russell's Logical Atomism

Published in print April 2012 | ISBN: 9780199651443
Published online September 2012 | e-ISBN: 9780191741197 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199651443.003.0006
Problems

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Russell’s attempt to obey the Vicious Circle Principle must lead to the conclusion that there are only countably many propositional functions, but this conflicts with his axiom of reducibility. So we do better to reject the ramified type theory, and thereby dispense with both. The simple type theory still has problems in deducing mathematics, e.g. over the axiom of infinity, and problems too over what this chapter calls ‘type-neutral’ predicates. A remedy for both of these points may be available, but it would take us into uncharted waters. One may observe that the axiom of choice (which Russell calls the ‘multiplicative’ axiom) is needed for the classical theory of infinite sets.

Keywords: Vicious Circle Principle; axiom of reducibility; ramified type theory; simple type theory; axiom of infinity, type-neutral predicates, axiom of choice; infinite sets

Chapter.  7697 words. 

Subjects: History of Western Philosophy

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