User blog comment:Ubersketch/Fundamental sequences for CK and higher?/@comment-35295276-20181025210550/@comment-35470197-20181025215252

In addition, Kleenes's O gives an explicit construction of a tree enumerating ordinals below \(\omega_1^{\textrm{CK}}\). (It is not computable, as you wrote.)

> all ordinals \(<\Omega\) have a fundamental sequence

Well, no. Just all (non-zero) limit ordinals \(< \Omega\) do.