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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