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Just another random mathematical speculation I came up with. It's probably not all that strong in terms of growth rate, but I thought it was interesting.
Define a 'hyperprime' number as a prime number whose digits, when added together, sum to a prime number, and the digits of that number added together also sum to a prime number, etc. until it reaches a 1  digit prime.
The basic Hyperprime counting function HP(x) = the xth hyperprime number after 1.
The recursive Hyperprime counting functon RHP(x) = HP(HP(HP(HP...(x)))...) iterated a number of times equal to the value of x. I could probably come up with more extentions but I'm tired.
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This is written as a series of numbers in parentheses separated by commas, for example (x, y, z).
The first number on the left (x) represents the input of a FGH function.
The second number (y) represents the yth function after a limit ordinal
The third number (z) represents the zth limit ordinal (in this case, if z is 0 it is treating 0 as a limit ordinal, even though it's really not one)
So (0, 1, 1) = F1(1)
(1, 1, 1) = fw+1(1)
(2, 2, 2) = fw2+2(2)
You can extend this with a fourth number, making (x, y, z, a), where a equals the ath fixed point after ω, meaning z is the zth limit ordinal after the fixed point defined by a, etc.
For a further extension, you can have (x, y, z, a, b), where b is the bth admissible ordinal after ω (if b is 0 you just…
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So I was thinking about Jonathan Bowers' Oblivion series numbers, and I came up with a similar idea. It's almost certainly just as undefined as they are, but I just wanted to post the idea and see if it was even theoretically workable.
First you start with the ChurchKleene Ordinal. Then, using the function of said Ordinal in the FastGrowing Hierarchy, you define the number f(CK)(10^100^100) (Function Googolplex of the ChurchKleene function). Call this number CK0. Then you define the simplest (meaning using the fewest different symbols) system of mathematics that has a prooftheoretic ordinal equal to the ChurchKleene Ordinal. Call this system SCK0.
Next you take the largest finite number that can be defined in SCK0 using no more than CK0…
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I know it's very vague, but as I'm only a novice Googologist, I just wanted to know if the general idea is plausible.
Step 1: Create a fastgrowing function that outputs not finite numbers, but ordinals
Step 2: Use this function to define a very large ordinal
Step 3: Create a fastgrowing function using the fundamental sequence of that ordinal
Step 4: Use this new function to define a very large finite number
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Can this be an official number on the wiki, with an article? It is named after me and my friend (Littlepeng9 didn't want any credit despite helping), since we helped come up with it. I am also considering naming it the EmlightenedPellucidArmstrong number, to reflect said poster'shelp. First we define the PPA (or EPA) Function as:
"The largest number defined by a formula φ, such that ZFC proves defines a unique number with a proof of length n"
And a proof of length n is defined as "n total applications of inference rules from the axioms of ZFC", using these rules and axiomization of ZFC: https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf
http://wwwmath.unimuenster.de/logik/Personen/Koulakis/Large_cardinals_and_elementary_embeddings_of_V.p…
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