impredicative definition meaning in Hindi

Examples

1. Poincar?and Weyl argued that impredicative definitions are problematic only when one or more underlying sets are infinite.
2. Set theories such as ZFC are not based on this sort of predicative framework, and allow impredicative definitions.
3. Then to exclude impredicative definitions within a type, the types above type 0 are further separated into orders.
4. But this separation into orders makes it impossible to construct the familiar analysis, which [ see Kleene's example at Impredicativity ] contains impredicative definitions.
5. This notion of collection-or or class-as-object, when used without restriction, results in Russell's paradox; see more below about impredicative definitions.
6. Kleene observes that " to exclude impredicative definitions within a type, the types above type 0 [ primary objects or individuals " not subjected to logical analysis " ] are further separated into orders.
7. If only definable properties are considered to exist, then the axiom means that to every impredicative definition within a given type there is an equivalent predicative one ( Kleene 1952 : 44 45 ).
8. He gives two examples of impredicative definitions & ndash; ( i ) the notion of Dedekind chains and ( ii ) " in analysis wherever the maximum or minimum of a previously defined " completed " set of numbers } } is used for further inferences.
9. Zermelo's example of minimum and maximum of a previously defined " completed " set of numbers reappears in Kleene 1952 : 42-42 where Kleene uses the example of Least upper bound in his discussion of impredicative definitions; Kleene does not resolve this problem.
10. G�del's approach to the law of excluded middle was to assert that objections against " the use of'impredicative definitions'" " carried more weight " than " the law of excluded middle and related theorems of the propositional calculus " ( Dawson p . 156 ).