Impredicativity

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In mathematics, impredicativity is the property of a self-referencing definition. More precisely, a definition is said to be impredicative if it depends on a set of things, at least one of which is the thing it defines.

Russell's paradox is a famous example of an impredicative construction: the set of all sets which do not contain themselves. The paradox is whether such a set contains itself or not — if it does then by definition it should not, and if it does not then by definition it should.

Nevertheless, the famous mathematician Frank P. Ramsey argued that impredicative definition is absolutely necessary. For instance, the definition of "Tallest person in the room" is impredicative, since it depends on a set of things of which it is an element, namely the set of all persons in the room. Concerning mathematics, an example of an impredicative definition is the smallest number in a set, which is formally defined as: y=min(X) if and only if For all elements x of X y is less than or equal to x.

The rejection of impredicatively specified objects (but the acceptance of the natural numbers as classically understood) leads to the position in the philosophy of mathematics known as Predicativism, taken by Henri Poincaré and (in Das Kontinuum) Hermann Weyl.