User blog comment:Mh314159/Help me understand a natural number recursion/@comment-24725252-20191018230855/@comment-39541634-20191019172413

Yeah. Many numbers map to the same ordinal in this system.

OTOH It looks like n=512 maps to w*2, n=4 maps to w^2 and n=5 maps to w^2+w. Other cases (n=13 for example) seem to correspond to much higher ordinals.So the system still seems to have considerable power.

(the numbers themselves follow the Hardy Hierarchy rather than the FGH, so w^2 is only equivalent to exponential growth rather than the w^2-level on the FGH)