Check out the original source of Sam's Number at Googology Wiki.
Member: 1 (this one!)
I made one of the flashiest signature number in my own googology history!
Initially created on February 7th, 2022.
I define the more well-defined version of the Sam’s number, since the original version of the Sam’s number is completely ill-defined and is not classified as an uncomputable number.
Here the definition of the more well-defined Sam’s number:
First, I define Sam(0) to be equal to Oblivion, and Sam(1) to be equal to Utter Oblivion.
Sam(n) is defined as "the largest finite number that can be uniquely defined using no more than Sam(n - 1) symbols". For example, Sam(2) is defined as "the largest finite number that can be uniquely defined using no more than Utter Oblivion = Sam(1) symbols"; Sam(3) is defined as "the largest finite number that can be uniquely defined using no more than Sam(2) symbols", and so on.
Sam[m+1](n) = Sam[m]^(Sam[m]^(...^Sam[m](n))(n))(n) where there are n Sam's.
Sam[A](n) = Sam[A[Sam(n)]](n) where A is a limit ordinal.
My Sam’s number is:
Sam[ω(ε(ω_1CK×f[ψ(K)×E100#100#100](Rayo(10^100))))CK](Large Number Garden Number)
That’s a salad number!
It is an ill-defined uncomputable number beyond Utter Oblivion. Its ill-definedness is simply Jonathan Bowers' Oblivion and Utter Oblivion are unformalized, although it is intended be larger than Rayo's number and Large Number Garden Number.
DECEMBER 25TH, 2025 UPDATE: MERRY CHRISTMAS! I created an even stronger version of Sam function called "Sam+ function", which diagonalizes over the unique Oblivion-based numbers from Numbers of the Feracity by simply changing the first two rules of the definition. The first two rules of the Sam+ function are:
Sam+(-1) is equal to True Cycles, Sam+(0) is equal to Uttery True Cycles, and Sam+(1) is equal to Dismentalize the Oblivion.
Sam+(n+1) is defined as "the largest finite number that can be uniquely defined using no more than Sam+(n) symbols with Sam+(n) recursion level n+1, using level n+1 recursion-based diagonalization that diagonalizes over the level n recursion stage from the precedent based in Numbers of the Feracity" with respect to the uncomputable fundamental sequence: Even More Devastrating Dismentalized Oblivion, Multiversal and Powerful Oblivious Explosions, Oblivious Fast-Fourier Transforms, Utterly High Five Oblivion Platform, Sixthful Approach, ...
The other two rules remain the same. The Sam+ function eventually dominates the limit of the regular Sam function, assuming that the Sam function is defined and there exists an ordinal S where f{S}(n) surpass any uncountable cardinals, including indescribable cardinal and rank-into-rank cardinal, since the Sam+ function includes the more complex cycle-based diagonalization that autonomously create an almost infinite loop, making the latter function far more powerful than the former one.
I also coined my stronger version of Sam's number called "Super Sam's number". It is defined as:
Sam+[ω(ε(ω_1CK×f[ψ(K)×E100#100#100](Rayo(10^100))))CK](Large Number Garden Number)
Super Sam's number isn't that much different in terms of the expression of the definition. The arguments in the function are pretty much the same, just swap the Sam function with the much more powerful Sam+ function.
Sam Number / Sam number
Dead Sam
Death Sam
Sammy Pranker
Sammy Hoaxer
Derpy Sam
Sammy Rayo
Ultra Sam
The Dead Jokers
S.A.M Number / S.A.M number
These alternative capitalizations are meant to serve as a joke with stronger irony. There are 2^10 = 1,024 different possible capitalizations, but here I picked up only eight (one has no 's in it).
SAM'S NUMBER
SAM NUMBER
SaM'S NuMBeR
sAm's nUmbEr
Sam's numbeR
SAm'S nuMBeR
SaM's NumbeR
Sam'S NumbeR