I would like to know whether the number of infinity sets of different cardinalities is uncountable or countable. Is the proof simple? Thanks.
Technically, neither one is true: the class of cardinalities is too large to form a set. (However, having to choose between "countable" and "uncountable", certainly "uncountable" seems better: there is at least an uncountable set of cardinalities of sets.)
I think this is equivalent to a previous question on this site, but anyway: please see Fact 20 of
and let me know if you have any further questions.
Contrary to the responses thus far (and with no disrespect intended to their authors), I think we can say quite unequivocally that there are strictly more than countably-many sets with distinct infinite cardinalities, and that (ipso facto) there are uncountably many such sets.
The answers before mine fixate on the idea of cardinality itself as the concept of quantity, and correctly point out that that there isn't a "set of infinite cardinalities". (There is a class of infinite cardinalities, if you like that sort of thing, but perhaps not everyone likes talking about proper classes.) Because there cannot be a set of infinite cardinalities, the reasoning goes, there cannot be a cardinality (a well-defined sense of "the number of") of infinite cardinalities. However, the original question isn't about "cardinality" per se: it's about a less formal notion of "the number of something", which we conventionally define by cardinality in the case of sets. As we are not talking about "the set of X", however, perhaps we shouldn't be using this definition of number. What then?
Well, the original question isn't asking us precisely how many infinite cardinalities there are: just whether this number is countable or uncountable --- a dichotomy of ordering; whether it is less-than-or-equal-to the "number" of natural numbers (i.e. whether it can be placed in bijective correspondance with a subset of the natural numbers), or not. This is, as far as I'm aware, the definition of the terms "countable" and "uncountable".
As the number of infinite cardinalities cannot be put into one-to-one correspondence with any subset of the natural numbers, it follows that the number of infinities is uncountable. The fact that they cannot be put into a set is irrelevant, in my opinion: at worst, one only needs to identify a single uncountable set U of infinities, note that there is no surjection onto it from the natural numbers, and note incidentally that there are more infinities outside of the set U anyway. Any cardinal ℵC (identified in the usual way with an ordinal), for C itself an uncountable ordinal, will suffice for U.
Thus, there are uncountably many different infinities, QED.
If by the number of cardinalities you mean the set of all cardinalities - then there is none.
The class of all cardinal numbers is a proper class, as it contains a copy of the class of ordinal numbers - which is not a set.
So to answer you question, it's not countable. It's so big it's not even a set.
For any set $X,$ the set $P(X)$ of all subsets of $X$ has a bigger cardinality than $X$ itself (for $X$ is finite this is easy, for $X$ infinite you need a clever argument from Cantor, obtaineble in any set theory text). You can do this again, obtaining:
card(X) < card(P(X)) < card(P(P(X))) < ...