User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35392788-20191030090941

Ah yes, the obvious problem of defining what a "reasonable fundamental sequence system" is. I'll say what I believe could be a good definition just after, but do you reckon that the statement can be true if made more rigorous ?

Anyway, a "reasonable FS system" should obey a few properties : - \(\alpha<\alpha[n]\) for all \(\alpha\) - \(\alpha[n]<\alpha[m]\iff n<m\) - \(\sup\limits_n \alpha[n] = \alpha\) - Most importantly, transfinite induction along \(\alpha\) is provable in some theory T iff \(\alpha[n]\) is provably total in T as well

I think these restrictions allow us to draw much more conclusive statements about using PTOs in the FGH.