Subcursion

Subcursion can be added to any function, in the simple way of a subscript.

e.g. q_3 (note that i will be using underscores to denote subscripts. I would like to be able to use mathjax or latex that isn't too hard to use, but sadly I cant find any.)

But wait, what does the _3 expand into?

I define RecRec(n,m) in the following recursive way

Variables n and m are within the set N where set N is the set of all whole numbers that are not negative.

RecRec(n,m) = [insert any function] if n= 0

RecRec(n,m) = RecRec(n-1,RecRec(n-1,RecRec(n-1,RecRec(n-1,m)))) if n >1.

So, for subcursion, I will create my version of RecRec, denoted R_sub(n,m)

I change the termination for RecRec(n,m) to this

RecRec(n,m) = q(m) (note that you can modify this function to be any one, and since i am using the q function, I'll use it like this. Just remember that q(n) represents the subcursed function.)

[]^n is basically repeat whatever is inside the brackets n times.

q_n(m) = [q^{R_sub(q_n-1(m),q_n-1(m))}_n-1(]^{q_n-1(m)}q_n-1(m))

^ stands for recursion, while _ is for subcursion.Note that i treat anything in the brackets as a function and recurse it q_n-1(m) times.

So, remember you can use subcursion on any function, and R_sub(n,m) will automatically set its base rule to be the subcursed function.

QGH

q is for quaiL

Let p be a large countable ordinal such to every limit ordinal m < p there is assigned a fundamental sequence (a strictly increasing sequence of ordinals whose supremum is m). A quick growing hierarchy of functions, q^m : N => N, for m<p is defined as the following:

q^0(n) = n+1

q^{m+1}(n) = {q^m}_n(n)

q^m(n) = q^{m[n]}(n) if m is a limit ordinal.

Note that in all these cases the ^ denotes the nth function in the hierarchy and not recursion.

We can also make other hierarchys in the way of quick iteration hierarchy.

Subcursion moment