impredicative

Sets which do not meet the requirement are called impredicative.

In the absence of it, the set of truth values is also considered impredicative.

In mathematics and logic, a self-referencing definition is called impredicative.

While these were fine for the situation, impredicative functions had to be disallowed:

He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions.

Then to exclude impredicative definitions within a type, the types above type 0 are further separated into orders.

The most general form of polymorphism is "higher-rank impredicative polymorphism".

This sort of definition of I was deemed by Poincaré to be "impredicative".

Russell's paradox is a famous example of an impredicative construction, namely the set of all sets which do not contain themselves.

For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals.

Π11-CA0 is stronger than arithmetical transfinite recursion and is fully impredicative.

This notion of collection-or or class-as-object, when used without restriction, results in Russell's paradox; see more below about impredicative definitions.

While IZF is based on constructive rather than classical logic, it is considered impredicative.

First inferences from the paradoxes [impredicative definitions, Logicism etc.], 13.

Arbitrary-rank and impredicative polymorphism.

Set theories such as ZFC are not based on this sort of predicative framework, and allow impredicative definitions.

In type theory, the most frequently studied impredicative typed λ-calculi are based on those of the lambda cube, especially System F.

Poincaré and Weyl argued that impredicative definitions are problematic only when one or more underlying sets are infinite.

This extension is not universally accepted by Intuitionists since it allows impredicative, i.e., circular, constructions, which are often identified with classical reasoning.

Wherein Zermelo rails against Poincaré's (and therefore Russell's) notion of impredicative definition.

Impredicative polymorphism (also called first-class polymorphism) is the most powerful form of parametric polymorphism.

(Once we have adopted an impredicative standpoint, abandoning the idea that classes are constructed, it is not unnatural to accept transfinite types.)

It is sometimes said to be the first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative".

Martin-Löf has modified his proposal a few times; his 1971 impredicative formulation was inconsistent as demonstrated by Girard's paradox.

More recently, predicativism has been studied by Solomon Feferman, who has used proof theory to explore the relationship between predicative and impredicative systems.