Hi,

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?)

MY THOUGHTS:

(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.