Impredicativity  

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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.
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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.

Something that is impredicative, in mathematics and logic, 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.

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Unless indicated otherwise, the text in this article is either based on Wikipedia article "Impredicativity" or another language Wikipedia page thereof used under the terms of the GNU Free Documentation License; or on original research by Jahsonic and friends. See Art and Popular Culture's copyright notice.

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