Difference between revisions of "Extended arrow notation"

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(Created page with "Extended arrow notation is a notation that was invented by Googology Wikia User Googleaarex: googology.wikia.com/wiki/User:Googleaarex.")
 
 
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Extended arrow notation is a notation that was invented by Googology Wikia User Googleaarex: [[googology.wikia.com/wiki/User:Googleaarex]].
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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 =
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Basic Notation is very simple. It generalizes the [[Knuth's up-arrow notation|normal arrow notation]].
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\(a \uparrow_2 b = a \underbrace{\uparrow\uparrow\dots\uparrow}_b a\)
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\(a \uparrow_3 b = a \underbrace{\uparrow_2\uparrow_2\dots\uparrow_2}_b a\)
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\(a \uparrow_n b = a \underbrace{\uparrow_{n-1}\uparrow_{n-1}\dots\uparrow_{n-1}}_b a\)
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Note that all parts of Extended arrow notation, like Knuth's up-arrow notation, have expressions that are evaluated from the right.
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Limit in FGH: \(f_\omega(n)\)
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= Nested up-arrow notation =
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To extend the notation here, we first have to make a change: \(\uparrow_n = \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_{n-1}}\)
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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 \(a \uparrow_{\uparrow\uparrow_2} b = a \underbrace{\uparrow_{\uparrow_2}\uparrow_{\uparrow_2}\dots\uparrow_{\uparrow_2}}_b a\)
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Then: \(a \uparrow_{\uparrow_{\uparrow_2}} b = a \uparrow_{\uparrow_{b+1}} a\) and so on.
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Limit: \(\varepsilon_0\)
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= Array up-arrow notation =
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Limit: \(\psi(\psi_{I(\omega, 0)}(0))\)
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= Dimensional array up-arrow notation =
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= Hyperarray up-arrow notation =
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= Legion array up-arrow notation =
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= Hyperlegion array up-arrow notation =

Latest revision as of 04:54, 8 November 2018

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

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

Dimensional array up-arrow notation

Hyperarray up-arrow notation

Legion array up-arrow notation

Hyperlegion array up-arrow notation