User:TachyonJack/Fun with large numbers

Fun with Large numbers
Using Conways Chained Arrow Notation

$$

F_0(n) \equiv 1

$$

$$

F_1(n) \equiv n

$$

$$ F_2(n) \equiv \begin{matrix}

\underbrace{n\rightarrow n\rightarrow \cdots \rightarrow n}\\ \ \ n \mbox{ copies of }n \end{matrix} $$

$$

F_3(n) \equiv \left. \begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right \} n \mbox{ layers}

$$

$$ F_4(n) \equiv \ \ \ \underbrace{ (n \mbox{ layers}) \left  \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

\left \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \end{matrix} \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix}\right. \Bigg \{ \cdots \cdots

\cdots

\cdots \cdots

\cdots \left \{ \begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \\ \vdots \end{matrix}  \qquad\;\;} \\

&\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

}_{\begin{matrix} n \mbox{ towers} \end{matrix} }

$$

$$ F_5(n) \equiv \ \ \left. \begin{matrix} &\underbrace{ n \mbox{ layers} \left  \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

\left \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \end{matrix} \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix}\right. \Bigg \{ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

\cdots \left \{ \begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \\ \vdots \end{matrix}  \qquad\;\;} \\

&\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

} \\ &\underbrace{\begin{matrix}\vdots \\ \vdots \\ \vdots \end{matrix}} \\ &\underbrace{ n \mbox{ layers} \left  \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

\left \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \end{matrix} \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix}\right. \Bigg \{ \cdots \cdots

\cdots \cdots

\cdots \cdots \cdots

\cdots \left \{ \begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \\ \vdots \end{matrix}  \qquad\;\;} \\

&\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

} \\ &\underbrace{ n \mbox{ layers} \left  \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

\left \{

\begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \end{matrix} \qquad\;\;} \\ &\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix}\right. \Bigg \{ \cdots \cdots \cdots

\cdots \left \{ \begin{matrix} &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \begin{matrix}\vdots \\ \vdots \\ \vdots \end{matrix}  \qquad\;\;} \\

&\underbrace{n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow n\rightarrow\cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right.

} \\ &\begin{matrix} n \mbox{ towers} \end{matrix} \end{matrix}\right \}n \mbox{ Super-Layers} $$ $$ G(n) \equiv F_n(n) $$