User blog comment:Mh314159/Alpha numbers (and beyond)/@comment-39585023-20191007101914/@comment-39585023-20191009233935

Forgive me, but I do not understand:

Your quoted definitions don't seem to match the ones I used. I did not define f(x) as a functional power of g(x), for example. And superscripting f(x) is more powerful than subscripting, because I defined the unsubscripted function to copy the argument to the subscript. So f2(2) is much larger than f2(2). This seems to contradict your comment unless I did not understand something.

If f(x) already has growth rate w+1, and if g(x) puts a functional power on f(x) that grows even faster than f(x), what is the growth rate? And h does to g what g did to f. And alpha(n,x)  extends the letters to numbers n that are themselves functions of letters. I have a hard time imagining that with all the times the argument gets copied to the subscript that alpha2(x) doesn't grow much faster than only w^2. The power on alpha is not just a numerical functional power, it is itself strong recursed in terms of alpha.