User blog:GamesFan2000/Explosively Expanding Function (Part 1: Finite Subscripts)

The Explosively Expanding Function is a new notation from me. This is meant to be an extremely fast-growing function. The rules of the basic form of the function are as follows:

Rule 1: The function is represented by EEa(n).

Rule 2: EEa(0) = 0, EEa(1) = 1

Rule 3: EE0(n) is the first non-trivial part of the function. Create an n-length chain of n’s with n right-arrows between each n. This will not follow Peter Hurford’s rules on multiple arrows between numbers, instead expanding out the right-most arrow groups first, solving the single-arrows up to the preceding multi-arrows and then applying the answer of the single-arrows to the preceding multi-arrows. Otherwise, the functionality is the same. Call the answer to the first expression a. Create an a-length chain of a’s with a right-arrows between each a. Call the answer of the second expression b. Repeat this logic n times. This has an approximate growth rate of ω^ω+1 on the FGH.

Example:

EE0(2): 2→22 = 4, 4→44→44→44 = 4→44→44→34→34→34 = 4→44→44→34→34→24→24→24 = 4→44→44→34→34→24→2(4→4→4→4)

Rule 4: EE1(n) is the next level of the equation. Take the answer of EE0(n) and nest it within an EE0(n)-long nested expression of EE0’s. This has an approximate growth rate of at the very least ω^ω^ω, and at most ε0, in the FGH. If someone can help me with a more exact growth rate, then that would be very helpful.

Example:

EE1(2)= EE0(EE0(…EE0(2)undefinedEE0’s…EE0((EE0(2)))…))

Rule 5: For any EEa(n) in which a is a finite number, the answer will be the expression built from the answer of EEa-1(n) nested in an EEa-1(n)-long nested expression of EEa-1’s. I would believe that this is well beyond ε0 in the FGH, possibly as large as ε1, but more likely in the range of ε0^ω and ε0^ε0. Again, if you can prove an exact growth rate, that would be appreciated.