User blog:Dchew89/General Idea for my d(n) Function

First definition of d(n) for ordinals: The largest ordinal that can be defined using only successorship, fundamental sequence climbing, and previously defined ordinals. Ordinals defined should only be represented using addition, multiplication, or exponentiation. d(0)=0

Fundamental sequence climbing: If an ordinal is the third (or second counting “0th, 1st, 2nd”) ordinal in a fundamental sequence from a system of defining fundamental sequences being used, then the supremum of that fundamental sequence may be defined as the next output.

Example: \(d(0)=0\space d(1)=1\space d(2)=2\space d(3)=\omega\space d(4)=\omega+1\space d(5)=\omega+2\space d(6)=\omega2\)

“The d(n) function for defining ordinals can only be used if given fundamental sequences or a general method for defining new fundamental sequences, otherwise having a limit of ⍵”

“The d(n) function should exhaust all fundamental sequences and methods for defining new fundamental sequences given a finite input, therefore having the limit of the largest ordinal definable with the given fundamental sequences exponentiated infinitely, assuming the first d(n) function definition.”

Beyond first definition: The d(n) function can be altered in its strength and speed either by changing what types of operators can be used to represent ordinals, such as hyperoperators vs the FGH, or by changing what fundamental sequences are used, such as veblen phi vs madore psi. Therefore, the d(n) function itself is not powerful, rather it represents the maximum strength of whatever it uses to define its outputs. This means that to pursue a truly powerful output with the d(n) function, creating a truly powerful system for defining fundamental sequences for ordinals would be one of the simplest strategies.

d(n) with FGH operators: It seems that due to the built in flexibility of the FGH with using countable ordinals of any size to define function growth rates, FGH operators may be useful for creating a very powerful d(n) function. However, even this function seems to be quite limited by the strength of nesting FGH subscripts, which, while incredibly powerful when used with small ordinals, becomes relatively obsolete, losing strength as larger ordinals are used for it.

Most of my work regarding the effects of using the FGH with transfinite ordinal inputs can be seen here, where the gray highlighted areas should be disregarded: https://docs.google.com/spreadsheets/d/1YaV1_mRyGtoFLXPC-NWMb-HbaemEyFIlO7Txf5DUXXE/edit?usp=sharing