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Paradox of the class of all grounded classes

Published online by Cambridge University Press:  12 March 2014

Shen Yuting
Shen Yuting*
Affiliation:
Tsing Hua University, Peking
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Extract

A class A for which there is an infinite progression of classes A1, A2, … (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 A1, A2, … such that

Since A1 ϵ K, A1 is a grounded class; since

A1 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 A1 is circular if there exists some positive integer n and classes A2, A3, …, An such that

For any given positive integer n, a class A1 is n-circular if there are classes A2, …, An, 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 18 , Issue 2 , June 1953 , pp. 114
Copyright
Copyright © Association for Symbolic Logic 1953

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References

1 The referee pointed out that on pp. 128–130 of Mathematical logic (revised ed., 1951, Cambridge, Mass., U.S.A.)Google Scholar, W. V. Quine proves a result (his *181) which amounts to the same thing as the paradox of the class of all classes which are not n-circular.