number of infinite sets with different cardinalities 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 

http://math.uga.edu/~pete/settheorypart1.pdf

and let me know if you have any further questions.
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) &lt; card(P(X)) &lt; card(P(P(X))) &lt; ...