Cardinality of the set of all ordinals.

[TylerH]

What would the cardinality of the set of all ordinal numbers be? Is it even known or does the question even make sense in the case of such a weird, almost paradoxical set?

 

[micromass]

The class of all ordinals doesn't form a set. There are simply too many ordinals to enclose them all in a set. Rather, the class of all ordinals forms a proper class. In ZFC set theory, there is no way to assign a cardinality to a proper class.

This should be interesting for you: http://en.wikipedia.org/wiki/Burali-Forti_paradox