User blog comment:Hyp cos/tree function and TREE(3)/@comment-5150073-20140624084600/@comment-5529393-20140625073752

@Ikosarakt1: I'm not sure why you say "proof-less". If we've proven that D(n) grows faster than any function provably recursive in higher order logic, and g(n) is provably recursive in higher order logic, then we've proven that D(n) grows faster than g(n). Perhaps you'd like an ordinal notation that goes up to an ordinal that represents the growth rate of D(n) - so would I, that would be very nice. But we don't need that to see that certain functions grow faster than certain other functions.

For TREE(n), we can associate each labelled tree with an ordinal such that no tree associated with a smaller ordinal will embed into a tree associated with a larger ordinal, and the ordinals go up to $$\vartheta(\Omega^{\omega} \omega$$.  So if we create a sequence of trees for TREE(m, n) by taking the tree associated with the largest ordinal smaller than all previous ordinals, and no more than the maximum allowed number of vertices, then the sequence will have length $$H_{\vartheta(\Omega^{\omega} \n} (m)$$. (assuming the correct definition of H)  So, if we can show that another function f grows no faster than $$H_\alpha (n)$$ for $$\alpha < \vartheta(\Omega^{\omega} \omega$$, then we have proven that TREE(n) grows faster than f.

The problem for functions like TREE(n) and SCG(n) is finding an upper bound. I believe proof theoretic techniques can be used to upper bound the functions, but that doesn't help in determining TREE(n) and SCG(n) for particular values of n.