User blog comment:Mh314159/FOX notation/@comment-39585023-20191205111828/@comment-39585023-20191206051211

OK, thanks. Since the subscripts were not helping me go beyond w^w, I will drop them from that set of rules and use them in another way. First I define [1] to be the function f and [a+1] to be the a-th function after f and I recurse [a+1] to [a] the same way I recursed g to f. Now I have an unlimited series of these functions, and here's what the subscript does.

[a]‹S›n(x) = [[a]‹S›n-1(x)]‹S›(x)

[a]‹S›0(x) = [a]‹S›(x)

to iterate the subscript, I use:

[a]‹S››α(n)(x) = [a]‹T›α(n-1)(x), α(j) = [a]‹T›α(j-1)(x), α(0) = [a]‹T›(x)

As before, T is S with the first term decremented and I will probably use the zero replacement rule we discussed, although you seemed to say that it doesn't make a big different to the ordinals

So for example, [a]‹S››α(9)(x) = [a]‹T›[a]‹T›...(x)(x) where the dots indicate 9 layers of subscript ending in [a]‹T›(x).

And of course, we can use huge n, such as [a]‹S››α([a]‹S›(x))(x) Now the subscripts diagonalize the sequence of functions and there are many layers of subscripts.

Perhaps a beta function could iterate layers of alpha functions, but I haven't worked that out yet.

Even if this isn't yet optimized, is it a good approach?