Circulation and the Fast-Growing Hierarchy

[Catullus]

\(|\alpha|\leq\aleph_0\) 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 ||.