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Impredicativity

In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions. The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over lower levels results in variables of some new type, distinguished from the lower types that the variable ranges over. A prototypical example is intuitionistic type theory, which retains ramification so as to discard impredicativity. Russell's paradox is a famous example of an impredicative construction—namely the set of all sets that do not contain themselves. The paradox is that such a set cannot exist: If it would exist, the question could be asked whether it contains itself or not — if it does then by definition it should not, and if it does not then by definition it should. The greatest lower bound of a set , glb(), also has an impredicative definition: = glb() if and only if for all elements of , is less than or equal to , and any less than or equal to all elements of is less than or equal to . This definition quantifies over the set (potentially infinite, depending on the order in question) whose members are the lower bounds of , one of which being the glb itself. Hence predicativism would reject this definition. The terms "predicative" and "impredicative" were introduced by , though the meaning has changed a little since then. Solomon Feferman provides a historical review of predicativity, connecting it to current outstanding research problems. The vicious circle principle was suggested by Henri Poincaré (1905-6, 1908) and Bertrand Russell in the wake of the paradoxes as a requirement on legitimate set specifications.

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Related concepts (16)
Axiom of reducibility
The axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types. Russell devised and introduced the axiom in an attempt to manage the contradictions he had discovered in his analysis of set theory. With Russell's discovery (1901, 1902) of a paradox in Gottlob Frege's 1879 Begriffsschrift and Frege's acknowledgment of the same (1902), Russell tentatively introduced his solution as "Appendix B: Doctrine of Types" in his 1903 The Principles of Mathematics.
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