Axiom of Reducibility

Discipline: Philosophy

Axiom introduced by English philosopher and mathematician Bertrand Russell (1872-1970) in connection with the ramified theory of types. It says that any higher-order property or proposition can be reduced to an equivalent first-order one.

The ramified theory of types caused difficulties for defining real numbers (using Dedekind sections) and for the process known as mathematical induction (roughly: if a property belongs to the first term in a series, and to the successor of any term to which it belongs, then it belongs to them all).

Russell introduced the axiom to deal with these problems, but it was widely felt to be unfounded, and was later dispensed with by Frank P. Ramsey (1903-1930) in Chapter 1 of his Foundations of Mathematics (1931).

Source:
B Russell, 'Mathematical Logic as Based on the Theory of Types', American Mathematical Monthly (1908); reprinted in R C Marsh, ed., Logic and Knowledge (1956) and in J van Heijenoort, ed., From Frege to Godel (1967)

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