impredicative construction

In mathematics, an impredicative construction defines an entity by referring to a collection that includes the entity itself. This self-referential approach, highlighted by mathematician Henri Poincaré, was seen as problematic because it assumes the existence of the entity before it is fully defined.1

For example, when defining the least upper bound of a set of real numbers, the definition might refer to properties of the bound itself, which is what is being defined. This contrasts with predicative constructions, which define entities without such self-reference.1

Bertrand Russell's paradox, which revealed a flaw in early set theory, stemmed from an impredicative definition of the set of all sets that are not members of themselves. While impredicative constructions are often necessary for classical analysis, mathematicians like Hermann Weyl and Solomon Feferman have shown they can sometimes be avoided. However, impredicative methods are highly useful in computer science for creating fixpoints.231