User blog comment:Fejfo/Super Fast Beaver Hierarchies and a weird OCF/@comment-35470197-20180806040506/@comment-35470197-20180806112708

> I don't know if I need to define it more formal than that. Because p is a sequence of instructions and I numbered the instructions the sequence can be ordered alphabetically (><[]~')

The problem is due to the first variable, i.e. the ordinal \(\alpha\). If you enumerate such pairs, then presenting \(L[i] = (\alpha_i,p_i)\), the limit \(\sup_{i \in \omega} \alpha_i\) is also a countable ordinal (not \(\Omega\) at all), which heavily depends on how you enumerate.

> I have thought about ways to fix this, I don't know if a more traditional encoding of ordinals as well-orderings would help but an infinite time version of bitf*ck might.

You do not have to encode ordinals into well-orderings on subsets of \(\omega\), as long as the definition of \((L[i])_{i < \omega}\) is fixed. If you had not skipped the way to fix it, then you would have noticed that the limit of your hierarchy is not \(\Omega\) but a certain countable ordinal.

> And is my usage of ωαCK correct?

I am sorry that I do not know so well about \(\omega_{\alpha}^{\textrm{CK}}\). I think that there are several (non-equivalent) formulations of \(\omega_{\alpha}^{\textrm{CK}}\) such as the one using the admissibility or \(\textrm{KP}\) set theory, you can formulate \(\omega_{\alpha}^{\textrm{CK}}\) in any reasonable way, as long as you write the definition. So if you say "I define \(\omega_{\alpha}^{\textrm{CK}}\) just as the ordinal corresponding to my \(\alpha\)-th hierarchy in my blog post", then it works in your blog posts.

> Seems to suggest an encoding with well-orderings doesn't work either could you explain this in more detail?

It is irrelevant to whether you encode ordinals into well-orderings on subsets of \(\omega\) that your hierarchy has a limit. On the other hand, if you do not give a precise definition of \(L[i]\) (especially the first entry \(\alpha_i\)), your hierarchy has not been defined yet, because you explicitly used the enumeration system \((L[i])_{i < \omega}\) in the definition of your hierarchy.