User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35470197-20191029224813

I am not an expert, and hence do not know specific facts on ACA_0. Therefore I modify the question so that I can answer. (Then it might not be helpful for you, though.)


 * Let T be a recursive theory including arithmetic. Assume that T proves the well-foundedness of each segment of an ordinal notation N. Fix a primitive recursive system of fundamental sequences on N. Let (t_n)_n be a sequence in N which gives its limit. Then how K(n) (defined in the same way for T) can be compared to E(n) (defined by using the fixed fundamental sequences (t_n)_n)?

I note that E(n) is the same as yours if N is the ordinal notation associated to Cantor normal form and (t_n)_n corresponds to (ω↑↑n)_n.

It is easy to show f_t is eventually dominated by K for any t∈N, because T proves the well-foundedness of the segment below t+1 and the resursive system on N of fundamental sequences associated to Wainer hierarchy is primitive recursive. In particular, f_{t_n} is eventually dominated by K for any n∈N.

If K(n) is smaller than f_{t_n}(n) for a sufficiently large n∈N, then it implies K is eventually dominated by E(n).

In general, it is difficult to precisely show the condition. Instead, we can consider the proof length L(n) of the totality of f_{t_n} with respect to N. If L(n) is significantly greater than 10^n, then it is natural to guess K(n) is eventually dominated by E(n).

The function L(n) is googologically approximated by the proof length of the well-foundedness of t_n with respect to N. On the other hand, the latter term tends to be itself a large number of level f_{t_n}(n) under the assumption that T does not proves the well-foundedness of N, i.e. PTO(T) = the ordinal type of N. In this case, L(n) is perhaps googologically approximated by E(n) itself, and then K(n) is perhaps eventually dominated by E(n). (Sorry for many guesses.)

Is your question derived from the estimation of finite promise game?