I've heard that there is no set that contains all the cardinal numbers {0, 1, 2, ... Aleph_0, Aleph_1, Aleph_2, ...}

For this to be true, obviously there must be infinite cardinals other than those of Aleph_n with n being a finite cardinal. Otherwise I just described the set here.

Is there a proof of this? I've looked it up and the best I've ever seen was that the cardinality of such a set would be bigger than any element in the set, but I fail to see why this has to be the case.