Thread: Cantor's Paradox

Cantor showed that there is no highest cardinal number. I've seen various sources that state without much argument that this entails that there is no set of all cardinal numbers. I was wondering why. What is the connection between these?

Originally Posted by Alan.Sutherland

Cantor showed that there is no highest cardinal number. I've seen various sources that state without much argument that this entails that there is no set of all cardinal numbers. I was wondering why. What is the connection between these?

Fact: For any set A, the set of all subsets of A has a cardinal number greater than that of A.

Let C be the set of all cardinals. By definition, then, card(C) is the greatest cardinal. But the cardinal number of the set of all subsets of C must be greater than card(C), a contradiction.

Originally Posted by AlephZero

Fact: For any set A, the set of all subsets of A has a cardinal number greater than that of A.

Let C be the set of all cardinals. By definition, then, card(C) is the greatest cardinal. But the cardinal number of the set of all subsets of C must be greater than card(C), a contradiction.

By definition card(C) is the number of cardinals, you will need to prove that it is also the greatest cardinal.

(for instance, if there were but a countable number of cardinals constructed by the power set process form the naturals, there would be no greatest cardinal, but card(C) would still be aleph_null, and the set of cardinals would contain a cardinal strictly greater than this)

CB