User blog comment:Mh314159/FOX notation/@comment-35470197-20191204015153/@comment-35470197-20191205034411

> I think it is similar to what you posted?

Right. Diagonalisation of f(x) is stronger than f<0,x,c>(x), but this change does not effect to ordinals. By a better similarity to FGH, people prefer f<0,x,c>, though.

> What about iterating? Could I use f<0,0,c+1>(x) = fp(x) and get faster growth for sufficiently large p?

Of course. But the iteration just effect to ordinals as +1 or something like that, and the limit does not change By a better similarity to FGH again, people prefer f<0,x,c> without iteration. (Personally, I often use the iteration because it can make a notation simpler in many cases.)

> How about f<0,0,c+1>(x) = fp(x) with strong p and where n is some iterated version of f<0,0,c>(x)?

It depends on n, but it usually doesn not effect to the limit. The limit is basically determined by the "order of the recursion" (how to choose functions for diagnalisations) instead of the "way to create a new strong function from previously defined functions". It is because the limit usually corresponds to an ordinal which is set theoretically closed under ω×　and +1.

> but that does not matter to the ultimate growth rate?

Right. It does not change the limit. Usually, the difference of the initial function does not effect to the limit by the same reason above.

> Should I be disappointed that all I have done with my notation is do almost exactly what the FGH already does?

Not at all. For example, even "completing a numerical notation by completely imitating FGH up to one's best ordinal" is difficult, and hence can be a good training. I mean, if one can create a notation whose limit is ω^ω^ω, then it can be quite difficult for him or her to create a notation completely the same as ω^ω^ω, because it requires good understanding of the recursion up to the limit.

Indeed, many of existsing notations are given by numerical notations which imitate FGH. It is not a non-sense work, because FGH iteself is not computable, i.e. is not defined in arithmetic with a numerical notation equipped with a decisive alogrithm and hence is belonging to uncomputable googology. In other word, we are not allowed to directly use FGH in computable googology, and hence we actually need non-trivial effort of "completely imitating" FGH without using ordinals.

> The subscript is a minor change? I thought it contributed a lot to the growth.

Sorry for disappointing you in this point, but honestly I should clarify that the rearrangement in that way does not effect the limit, because you have already achieved at ω^ω. To be more precise, when you were at ω^2-level, the subscript gave you a great development, because the change causes the replacement of +1 by +ω, which causes ω× for the limit. So a notation of ω^2-level becomes a notation of ω^3-level. The ordinal ω^ω is the least wall, for which ω× does not effect the limit strength. Stronger notations you create, stronger strategies you will need.

> Would it be more powerful if I combined it with iteration, like f<(a,b,c)>(x) = f<(a-1,b,c)>xp(x) ?

This change causes the replacement of +1 by +ω+1, and hence causes (ω+1)× for the limit. Since (ω+1)×ω^ω = ω^ω, the change does not effect to the limit strength. In other words, ω^ω, which you have achived, is so big that these methods do ot change the strength.

>I kind of feel a little disappointed that all I have done is rephrase the FGH. I want to do something original.

I should confirm you that "as powerful as FGH" does not mean "not original". Since FGH is defined in set theory, which are much more powerful than arithmetic, many arithmetic notations can be roughly approximated by FGH.

Say, we have a list of the highest computable googolism. You might be surprising that except for numbers defined by Hyperfactorial array notation and ill-defined numbers, almost all numbers are defined by a notation imitating FGH. So FGH can be regarded as a general framework to encode an airthmetic notation equipped with data of "how to choose functions for diagonalisations" into an actual function, and is not a single notation. (Actually, it is not computable, and is not a notation.) You can regard it as kind of a category of strategies.

So your strategy in the future might be categorised as a notation similar to FGH, but it does not mean that the notation is not original.