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Paradox of the class of all grounded classes

1953; Cambridge University Press; Volume: 18; Issue: 2 Linguagem: Inglês

10.2307/2268942

1943-5886

Yu-Ting Shen,

Rings, Modules, and Algebras

A class A for which there is an infinite progression of classes A 1 , A 2 , … (not necessarily all distinct) such that is said to be groundless. A class which is not groundless is said to be grounded. Let K be the class of all grounded classes. Let us assume that K is a groundless class. Then there is an infinite progression of classes A 1 , A 2 , … such that Since A 1 ϵ K , A 1 is a grounded class; since A 1 is also a groundless class. But this is impossible. Therefore K is a grounded class. Hence K ϵ K , and we have Therefore K is also a groundless class. This paradox forms a sort of triplet with the paradox of the class of all non-circular classes and the paradox of the class of all classes which are not n -circular ( n a given natural number). The last of the three includes as a special case the paradox of the class of all classes which are not members of themselves ( n = 1). More exactly, a class A 1 is circular if there exists some positive integer n and classes A 2 , A 3 , …, A n such that For any given positive integer n , a class A 1 is n -circular if there are classes A 2 , …, A n , such that Quite obviously, by arguments similar to the above, we get a paradox of the class of all non-circular classes and a paradox of the class of all classes which are not n -circular, for each positive integer n .