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Please do not use "..." when defining notations. Ellipses are helpful for giving a visual demonstration of how an expression expands, but they are often unclear or ambiguous, especially for complex recursion as seen in array notations. For beginners, I recommend that the "..." symbol not be used at all except in examples.
Use recursion instead. Instead of writing
 {a,b,c+1} = a {c} a {c} ... (b times) ... {c} a {c} a
you should write
 {a,1,c} = a
 {a,b+1,c+1} = {a,{a,b,c+1},c}.

In this post, I'm going to demonstrate an alternative to Kleene's O that might be easier to understand. It assumes some familiarity with programming.
Kleene's JavaScript is a textual notation for notating any recursive ordinal, defined like so:
 "0" is a Kleene's JavaScript program. It evaluates to the ordinal 0.
 Let P be a Kleene's JavaScript program that evaluates to ordinal a. Then if you append "+1" to P, the result is a Kleene's JavaScript program that evaluates to the ordinal a + 1.
 Let F be JavaScript code that consists of anonymous
function f:
 f takes a single nonnegative integer as an argument. (This is a theoretical variant of JS where integers can be infinitely large.)
 f is deterministic.
 f returns a string for every valid argument. That stringâ€¦

I formed the Great Googology Census 2 after having some philosophical differences with Sbiis Saibian's Great Googology Census. I feel many valuable contributions to googology have been snubbed. Without further ado, here they are.
Googology Certificate  Alex Thurber
Alex Thurber was a kid I knew in fifth grade who once said "I already told you a borgillion times." This had a huge impact on the googology community, and laid down the foundations for NumbersGuy1999's sequence trorgillion, quadrorgillion, etc. To my knowledge, Alex Thurber is currently in jail.
Googology Certificate  StarWarsGavin3
StarWarsGavin3 is best known for his pioneering work in creating enormous sprawling tables of fastgrowing hierarchy approximations.
\begin{eqnarray*} â€¦
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Hey everyone, I've been interested in large numbers for a long time and I am really excited to discover this wiki for the first time. After I saw all these mindblowing fastgrowing functions on your site I just HAD to create my own! :D So would you mind having a look at my notation? It's called SuperLambda Array Notation, a new array notation I invented that I think goes beyond the power of ExE, BEAF, and HAN. I'm kinda new at this googology stuff, so feedback is welcome :P In making this notation I tried to start from the ground up and find ways to strengthen the notation on every level. The notation has the following rules:
 Every expression must start with Î› (capital Greek lambda).
 Expressions can contain any list of numbers.
 Î›n = n^n
 Î›#,1â€¦

The proof that \(\Sigma >^*\) all computable functions is nontrivial but straightforward. I thought I would share it.
Theorem. \(\Sigma >^* f\) for all computable \(f : \mathbb{N} \rightarrow \mathbb{N}\).
Proof. We first establish a convention that the inputs and outputs to our Turing machines are nonnegative integers in the form of consecutive blocks of ones in otherwise blank tapes. The Turing machine must start on the rightmost one in input, and it must halt on the rightmost one in output.
Given a computable \(f\), define \(F\) as
\[F(n) = \sum_{i = 0}^n (f(i) + i^2).\]
We see that \(F(n) \geq f(n)\) and \(F(n) \geq n^2\), and that \(F\) is computable and increasing. Therefore there is a Turing machine \(M_F\) that computes \(F\) â€” say it hasâ€¦
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