|Impredicativeness: Impredicatives are concepts which are defined only by means of the propositional sets to which they themselves belong. Problems arise in connection with possible circular conclusions. To avoid paradoxes, the demand is sometimes made to avoid impredicative concepts. See also Paradoxes, Russellian Paradoxy, Poincaré.|
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Definition impredicative/Field: completely impredicative properties: are not at all derived from previously available properties. - In particular, there is no property to be a property. - Quasi-impredicative: also allows "property to be a property".
Classic example for impredicative definition: E.g. What is it for an ordinal number to be finite? - Fin (OZ) P [P is inductive & P (0)> P (OZ)] - whereby P is inductive is defined as:
b [P(b) > P(b + 1)] - ((s) All successors have the same property (to be a number)). - The invalid objection against the impredicative definition (VsImpredicativity) is that one cannot know that a given number, e.g. 2 is finite because, in order to show this, we must be able to show that 2 has every inductive property of 0 - to show that 2 is finite, we must show first that exactly this 2 is finite (circular). - Solution/Field: the solution is simple: if finiteness is an inductive property, then 2 is finite. - No circle.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980