Circulation and the Fast-Growing Hierarchy

[JmsNxn]

\(|\alpha|\leq\aleph_0\) does not hold for all countable ordinals. 

Yes it does, by definition of countable.

\(|\alpha|\leq\aleph_1\) does hold for all countable ordinals.

Nope it doesn't. By definition of uncountable and of \(\omega_1=\aleph_1\).
Conterproof. let \(\alpha=\omega_1\), then \(|\alpha|\leq\aleph_1\) is true but \(\alpha\) is uncountable.

Oh, you don't mean absolute value by ||.

What do you mean? I thought you meant cardinality, as well.

It would help if you went further into depth in your responses, Catullus. Not trying to sound snobbish or anything, it just helps with legibility.