Impredicativity
In mathematics and logic, a self-referencing definition is called
impredicative. More precisely, a definition is said to be
impredicative if it invokes the set being defined, or another set which contains the thing being defined. The opposite of impredicativity is predicativity, which essentially entails building stratified 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, also has an impredicative definition;
y = glb 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.