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.
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