User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35470197-20191031010724

@Syst3ms

The "only if" part of the fourth condition, i.e. "the well-foundedness of \(\alpha\) (with respect to a fixed ordinal notation so that the statement makes sense in arithmetic) implies the totality of the map \(n \mapsto \alpha[n]\)" does not have any issue, because we often assume the primitive recursiveness of the system of fundamental sequences. So we can assume the revised fourth condition.

The point is that even if we assume the revised four conditions, the following statement is false:
 * For any recursive theory \(T\), any ordinal notation \(S\), and any recursive system of fundamental sequences on \(S\) satisfying the first three conditions and the provable totality under \(T\) (which implies the revised fourth condition), if the ordinal type of \(S\) is greater than \(\textrm{PTO}(T)\), then \(f_{\textrm{PTO}(T)}\) with respect to \(S\) and the fixed system of fundamental sequences dominates any computable function provably total under \(T\).

The reason is that FGH can break, i.e. given ordinals \(\alpha < \beta\) corresponding to terms in \(S\), \(f_{\alpha}\) is not necessarily eventually dominated by \(f_{\beta}\). In order to avoid such phenomenon, we need to assume an additional non-trivial condition, which I am not certain.

As we know, FGH is not a canonical choice from the viewpoint of proof theory. Therefore we need to assume a reasonable condition which connects the growth rate of FGH and growth rates of provably total computable functions.

Of course, the majority of googologists believe that "well-known" fundamental sequences work well, as long as they are actually published ones. On the other hand, when we talk about unpublished (or imaginary) fundamental sequences such as those for Rathjen's ordinal notations and TON, we have no intuitive justification of the order-preserving property "^\(\alpha < \beta \to f_{\alpha} < f_{\beta}\)". (I do not mean that I jave a justification for a lower level, though.)