User blog:Bubby3/Letter notation extension to R and higher letters

Previously, I posted an extension of letter notation to Q here. Here is my extension of letter notation to R based on other letters. It is designed to put the previous letters in a more general form, and make the notation more extensible to as high as you want, just if you have fundemental sequences for the ordinals you are dealing with. I first posted the definition on the Googology discord server  which can be found if you search 8ZJ303DZM35H3ANBGXV6.

Definition of the notation

 * (a)|n = (a[n])|10, if n>= 2 and n is an integer (where n denotes the nth term of the fundemental sequence of a, defined here )
 * (a)|n = ((a[int(n)]~a[int(n)+1])[n])|10
 * (a)|n = (a[2])|(10^(n-1)) for n =<1<2
 * (a)|n = 10^n for n < 1
 * Rn = (\(\gamma_0\))|n

Definition of (a~b)|n
When I included this notation in the definition, you wonder what it is. It is a notation to enurmate ordinals betwen a and b[10], and (a~b)|n denotes an ordinal n of the way between a and b[10], and a and b are ordinals, and n is a number between 0 and 1.

It is defined as (a~b)[n] = (b{int(c)}~b{frac(c)+1})[frac(c)] when b is a limit, where c and a{n} defined as follows We also have to define the sucessor for when the ordinal, which is: Unfortanetly, this results in P10 = Q3, so we have to shift Q up by one, and Qn = Kn for n<2, with the fundemental sequence we chose, but this allows far more extensiblity, as long as fundemental sequences are consistent.
 * If there exists an m where b[m] = a
 * If m = 0, then c = 10*n and b{d} is b[d]
 * If m > 0, then c = (10/m)^n*m and b{d} is b[d]
 * (1)|n = 10^n
 * a|(b+1)|n = a|(b)(...a|(b)|(10^frac(n))) where a|(b) is applied int(n) times
 * Otherwise, then
 * If b < a[1], then c = 10*n and b{0} = a and b[d] when d is not 0
 * If b > a[1], then let p be the least nonegatative p such that b[p] > a, let c be (p * (10/p)^n) and b{p} = a and b[d] when d is not p.
 * a|((b~(b+1))[n])|10 = (b+1)|(2*5^n)