Why is that a harder question? It's a direct corollary of Cantor's Theorem that there is no largest cardinal number (assuming the powerset axiom, of course).
Like I said in my response to pndmnm, in my view if someone "[would] think that the cardinals also formed a set with an (infinite) cardinality of its own" then they haven't really grasped the theorem yet. It's built into it that if we're always allowed to form the powerset of a set then regardless of how large a cardinal you can find, you can always take its powerset and obtain a larger cardinal.
It's sort of interesting -- there's also no largest natural number, but we can talk about the size of their cardinality. You can't do that with all of the "infinities" since the collection of ordinals doesn't form an ordinal (Burali-Forti) and you can't form the set of all cardinals.
I'm not saying it's not interesting—in fact, I think it's fascinating—but all of this is implicit in Cantor's Theorem. "Harder question" to me implies there's something there that goes beyond the fundamental result that the powerset of a set X has strictly larger cardinality than X. One might think that it's just harder to understand, but I would dispute that too: if someone thinks they get Cantor's Theorem but has trouble grasping that there is no set of all sets, or largest cardinal number, it just shows that they don't really understand the theorem itself yet.
qntm|13 years ago
Likewise, there's no largest cardinal, but you'd think that the cardinals also formed a set with an (infinite) cardinality of its own.
Fun fact: they don't. There are too many of them.
ionfish|13 years ago
pndmnm|13 years ago
ionfish|13 years ago