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This my extended Rayo's notation (this is meant to be an extension of Rayo's).

Base

Let's say we have "x". We're going to set x to ((10^{100}\uparrow^{10^{100}}10^{100})\&_{10^{100}}10^{100}) using BEAF notation. Now that we have x we're going to do this (x![\underbrace{x,x,...,x}_{\text{x times}}])?. This is a Superplex (S or S_{0})

Small Uncomputable Numbers (SUN's)

For any small (not really small) uncomputable numbers- we don't use an array notation. Instead we say, S_{n}=\underbrace{Rayo_{S_{n-1}}(Rayo_{S_{n-1}}(...Rayo_{S_{n-1}}(n)))}_{\text{n times}}.

Names

S_{0} is a Superplex
S_{1} is a Super-duperplex
S_{2} is a Super-two-duperplex
.
.
.
S_{n} is a Super-"n"-duperplex

Mediocre Uncomputable Numbers (MUN's)

Now we have our mediocre uncomputable numbers- we use an array notation. Now we say S_{n,m}=\underbrace{S_{S_{._{._{._{S_{n}}}}}}}_{\text{m times}} (alternatively S[n,m]). S_{10^{100},10^{100}} is a Hyperplex.

Large Uncomputable Numbers (LUN's)

Next are Large Uncomputable Numbers. S_{n,m,p}=\underbrace{S_{n,S_{n,_{._{._{._{S_{n,m}}}}}}}}_{\text{p times}}. This pattern of repetition follows, so S_{n,m,p,q}=\underbrace{S_{n,m,S_{n,m,_{._{._{._{S_{n,m,p}}}}}}}}_{\text{q times}}.

Beyond Uncomputable Numbers (BUN's)

These numbers go beyond what the normal array function can show.

Complexed Numbers (CPN's)

Complexed Numbers can be are the smallest of the BUN's. They are represented by C[n,m]=S\underbrace{[n,n,...n]}_{\text{m times}}.

Complexed Numbers can also be represented by C^{p}[n,m]=C^{p-1}[n,C[n,m]]. C^{\text{Hyperplex}}[\text{Hyperplex},\text{Hyperplex}] is a Supercomplex.

Totally Uncomputable Numbers (TUN's)

Totally Uncomputable Numbers are Numbers that cannot be represented by Complexed Numbers. They are represented by U^{p}[n,m]. U^{1}[n,m]=\underbrace{C^{C^{.^{.^{.^{C^{1}[n,n]}}}}[n,n]}[n,n]}_{\text{m times}}, and U^{p}[n,m]=\underbrace{U^{U^{.^{.^{.^{U^{1}[n,m]}}}}[n,m]}[n,m]}_{\text{p times}}.

Infiniplex Function

A Infiniplex (I call it that because it might as well be infinity) is I_{0}=U^{\text{Supercomplex}}[\text{Supercomplex},\text{Supercomplex}].
I_{n}=\underbrace{U^{U^{.^{.^{.^{U^{1}[I_{0},I_{0}]}}}}I_{0},I_{0}]}[I_{0},I_{0}]}_{I_{n-1} \text{times}}.
\underbrace{I_{I_{._{._{._{0}}}}}}_{I_{0} \text{ times}} is an Infinihyperplex.


Names for Numbers Larger than an Infiniplex

Omegaplex I_{\omega}
Infinihyperplex \underbrace{I_{I_{._{._{._{0}}}}}}_{I_{0} \text{ times}}
Infinimega \underbrace{I_{I_{._{._{._{0}}}}}}_{I_{\omega} \text{ times}}
Omagahyperplex \underbrace{I_{\omega^{I_{\omega^{.^{.^{.^{I_\omega}}}}}}}}_{I_{\omega} \text{ times}}

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