|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.|
Books on Amazon
|Thiel I 324
Impredicativeness/Paradoxes/Poincaré: Poincaré believed with this that the decisive criterion had been found: illegitimate, "non-predicative" conditions are those that contain such a circle. ( > impredicative, Russell).
At first, it seemed sufficient to require expressions for the relation between element and set that in "x y" the second relation term y should belong to exactly one step higher than x (simple > type theory), thus the requirement that each permissible expression should be formed not only "predicatively" itself (i.e. not impredicatively) but also all arguments occurring in it must meet this condition, to form a "> branched type theory".
Philosophie und Mathematik Darmstadt 1995