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Russell's Logical Atomism$
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David Bostock

Print publication date: 2012

Print ISBN-13: 9780199651443

Published to Oxford Scholarship Online: September 2012

DOI: 10.1093/acprof:oso/9780199651443.001.0001

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(p.97) 6 Problems
Russell's Logical Atomism

David Bostock

Oxford University Press

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

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