User blog comment:Hyp cos/tree function and TREE(3)/@comment-5150073-20140624084600/@comment-5150073-20140625055016

FBZ, I meant for fixed n, so it can't increase and stay at certain recursive ordinal and represents computable function.

And yes, there is a problem when we evaluate $$f_{\omega_1^\text{CK}[65536]}(m)$$, a problem that it is unclear how it compares to all previous functions. We can say that it is certainly fast, extremely fast-growing, but that's proof-less. So, we just haven't enough examples showing its strength. The same thing goes to TREE(n), SCG(n) and mostly to D(n) and Friedman's promise games. The fact that they outgrow anything in a certain proof-theoretic system isn't so notable in this sense, because these systems don't propose notations for all these ordinals. Namely notations are useful.

Hyp cos, I can't be sure that this sequence even cannot be infinite. What if tree(n) is Turing complete?