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

What is the precise definition of the notion of a "program" in this blog post? How do you enumerate all programs alphabetically? In order to enusure that your hierarchy is well-defined, you need to give an explicit way to define the \(i\)-th program \(L[i]\) for each natural number \(i\).

If you define the \(i\)-th program \(L[i]\) as a pair \((\alpha_i,p_i)\) of an ordinal \(\alpha_i\) and other data \(p_i\), then \(\{\alpha_i \mid i \in \omega\}\) is automatically bounded by \(\omega_1\). It never goes beyong uncountables by definition. So then, your hierarchy is not well-defined for uncountable ordinals at all.

It is easy to enumerate Turing machines in arithmetic, but is difficult to enumerate Turing machines with transfinitely higher oracles in any reasonable theory.