The hierarchy of terminating recursive programs over N

Accessible Segments of the Fast Growing Hierarchy

“…Either one simply accepts this situation and does the best one can (and we will, later, in section 4); or alternatively one could shift the goalposts and try to reconsider the problem in a more amenable, generalized setting. One suitable place to look is the theory of type two recursive functionals where, as noted by Kleene (1958) and in stark contrast with the recursive functions, a good notation-free hierarchy already exists -by classifying total recursive functionals according to the ordinal heights of their trees of unsecured sequences (see also Wainer (1995)). This opens the possibility of coding and comparing recursive ordinals in terms of the majorization relationship between certain descent-recursive functionals which represent them.…”