User blog comment:B1mb0w/Strong D Function/@comment-1605058-20150624061803/@comment-1605058-20150702084349

That I connect an ordinal to every instance of \(D(x_1,...,x_n)\) has nothing to do with any assumptions on strength of the function or size of resulting numbers.

I could, indeed, do similar trick with TREE(n) - I could, to every labelled tree, relate an ordinal smaller than, say, \(\omega^\omega\). Indeed, I could to every tree relate a finite natural number, simply by enumerating all trees. The difference in this case would be that transfinite induction wouldn't work in that case, because we have no guarantee that next tree will lead to smaller ordinal.

In case of your function, I have decided to use this relation between instances of D function and ordinals below \(\omega^\omega\). In order to do that I made absolutely no assumptions on strength. It only later turned out that this relation makes transfinite induction work.