Is the collection of all cardinalities a set or a proper class?

Question

Is the collection of all cardinalities a set or a proper class? Does anybody ever think about the problem?

Answer 1

Collection of all cardinalities is indeed a proper class. To see this, note that there is at least as many cardinal numbers as ordinal numbers, because map $\alpha\rightarrow\aleph_\alpha$ is an injection.

Answer 2

The collection of all cardinal numbers is a proper class. If it were a set $X$, then $(\sup X)^+$ would be a cardinal number greater than every element of $X$, which is a contradiction.

(The supremum of a set of cardinals is given by its union, and the cardinal successor $\kappa^+$ of a cardinal $\kappa$ is defined as the Hartogs number of $\kappa$.)

Answer 3

It is a proper class. There are several ways to see this. One is: suppose $X$ were the set of all cardinals (=initial ordinals). The ordinals - in particular, the cardinals - are well-ordered, so we may add them together (indexed by this well-order) to form a single "super-cardinal;" but it's easy to check that this super-cardinal is larger than any element of $X$.

This is essentially the same reasoning as the Burali-Forti paradox http://en.wikipedia.org/wiki/Burali-Forti_paradox, which shows that the class of ordinals is a proper class.