User blog comment:Bubby3/Church Kleene ordinal question/@comment-1605058-20170810162247

With access to the halting oracle, you still cannot compute ordinals greater than $$\omega_1^\text{CK}$$. The set of computable ordinals is, in fact, still the same. However, given an oracle for $$\omega_1^\text{CK}$$, we can compute larger ordinals, and in fact the ordinals computable from $$\omega_1^\text{CK}$$ are exactly the ones below $$\omega_2^\text{CK}$$.

The part about normal functions is incorrect - the supremum of F(0) ranging over normal functions doesn't exist, since $$F(\alpha)=\beta+\alpha$$ is normal for every $$\beta$$.