User blog comment:Mh314159/Help me understand a natural number recursion/@comment-35470197-20191014142325/@comment-35470197-20191016115511

@Plain'N'Simple

> (the second bound is due to the fact that the termination of TREE(n) can be proven in a certain theory with a PTO of φ(ω@ω)).

Thank you for the information. But I could not precisely understand your statement. Do you mean that the provability of the termination of a function (or variants of Kruskal's theorem in this case) in some theory with PTO of φ(ω@ω) implies that the function is bounded by φ(ω@ω) with respect to the fundamental sequence associated to Veblen hierarchy, right? Then is there a source of the statement?

As you might know, f_α in FGH does not necessarily eventually dominate total computable functions whose termination is provable in a theory in which the well-foundedness of (an ordinal notation whose limit is greater than or equal to) α. Therefore your statement is not trivial for me.

I note that the estimation of TREE by Friedman using SCG is not given in that way. Friedman directly studied the relation between the halting time of Turing machines and proof length in Π_1^1-CA_0.