I know that there are "infinitely many" degrees of infinity. The steps for the proof I have seen:
1. For any cardinal, k, k<2^k
2. So we have countably many different degrees of infinity by considering; (aleph_null)<2^(aleph_null)<2^(2^(aleph_null))<...
I was wondering if it is know how many degrees of infinity there are (eg are there only countably many, uncountably many, even more?)
(Assuming AC) The cardinals are identified with the ordinals so this is the same question as asking how many ordinals there are. But the ordinals form a proper class (a class which is not a set) so can we even find a cardinality for the ordinals?
Thanks in advance for any replies.