# Difference between revisions of "Extended arrow notation"

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− | Extended arrow notation is a notation that was invented by Googology Wikia User Googleaarex: [ | + | Extended arrow notation is a notation that was invented by Googology Wikia User Googleaarex: [https://googology.wikia.com/wiki/User:Googleaarex/]. |

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+ | = Basic Notation = | ||

+ | |||

+ | Basic Notation is very simple. It generalizes the [[Knuth's up-arrow notation|normal arrow notation]]. | ||

+ | |||

+ | \(a \uparrow_2 b = a \underbrace{\uparrow\uparrow\dots\uparrow}_b a\) | ||

+ | |||

+ | \(a \uparrow_3 b = a \underbrace{\uparrow_2\uparrow_2\dots\uparrow_2}_b a\) | ||

+ | |||

+ | \(a \uparrow_n b = a \underbrace{\uparrow_{n-1}\uparrow_{n-1}\dots\uparrow_{n-1}}_b a\) | ||

+ | |||

+ | Note that all parts of Extended arrow notation, like Knuth's up-arrow notation, have expressions that are evaluated from the right. | ||

+ | |||

+ | Limit in FGH: \(f_\omega(n)\) | ||

+ | |||

+ | = Nested up-arrow notation = | ||

+ | |||

+ | To extend the notation here, we first have to make a change: \(\uparrow_n = \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_{n-1}}\) | ||

+ | |||

+ | Then we turn the problem into Basic notation: \(a \uparrow_{\uparrow_2} b = a \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_b} a = a \uparrow_{b+1} b\), and \(a \uparrow_{\uparrow\uparrow_2} b = a \underbrace{\uparrow_{\uparrow_2}\uparrow_{\uparrow_2}\dots\uparrow_{\uparrow_2}}_b a\) | ||

+ | |||

+ | Then: \(a \uparrow_{\uparrow_{\uparrow_2}} b = a \uparrow_{\uparrow_{b+1}} a\) and so on. | ||

+ | |||

+ | Limit: \(\varepsilon_0\) | ||

+ | |||

+ | = Array up-arrow notation = | ||

+ | |||

+ | Limit: \(\psi(\psi_{I(\omega, 0)}(0))\) | ||

+ | |||

+ | = Dimensional array up-arrow notation = | ||

+ | |||

+ | = Hyperarray up-arrow notation = | ||

+ | |||

+ | = Legion array up-arrow notation = | ||

+ | |||

+ | = Hyperlegion array up-arrow notation = |

## Revision as of 05:54, 8 November 2018

Extended arrow notation is a notation that was invented by Googology Wikia User Googleaarex: [1].

## Contents

# Basic Notation

Basic Notation is very simple. It generalizes the normal arrow notation.

\(a \uparrow_2 b = a \underbrace{\uparrow\uparrow\dots\uparrow}_b a\)

\(a \uparrow_3 b = a \underbrace{\uparrow_2\uparrow_2\dots\uparrow_2}_b a\)

\(a \uparrow_n b = a \underbrace{\uparrow_{n-1}\uparrow_{n-1}\dots\uparrow_{n-1}}_b a\)

Note that all parts of Extended arrow notation, like Knuth's up-arrow notation, have expressions that are evaluated from the right.

Limit in FGH: \(f_\omega(n)\)

# Nested up-arrow notation

To extend the notation here, we first have to make a change: \(\uparrow_n = \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_{n-1}}\)

Then we turn the problem into Basic notation: \(a \uparrow_{\uparrow_2} b = a \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_b} a = a \uparrow_{b+1} b\), and \(a \uparrow_{\uparrow\uparrow_2} b = a \underbrace{\uparrow_{\uparrow_2}\uparrow_{\uparrow_2}\dots\uparrow_{\uparrow_2}}_b a\)

Then: \(a \uparrow_{\uparrow_{\uparrow_2}} b = a \uparrow_{\uparrow_{b+1}} a\) and so on.

Limit: \(\varepsilon_0\)

# Array up-arrow notation

Limit: \(\psi(\psi_{I(\omega, 0)}(0))\)