Impredicativity

In mathematics and logic, a self-referencing definition is called impredicative. Roughly speaking, a definition is said to be impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set which contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative: many different authors have given different but related definitions of what the words mean.

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 but discards impredicativity.

Russell's paradox is a famous example of an impredicative construction, namely 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.

The greatest lower bound of a set X, glb(X), also has an impredicative definition: y = glb(X) if and only if for all elements x of X, y is less than or equal to x, and any z less than or equal to all elements of X is less than or equal to y. But this definition also quantifies over the set (potentially infinite, depending on the order in question) whose members are the lower bounds of X, one of which being the glb itself. Hence predicativism would reject this definition.

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