Difference between revisions of "Extended arrow notation"

From Cantor's Attic
Jump to: navigation, search
Line 12: Line 12:
  
 
Note that all parts of Extended arrow notation, like Knuth's up-arrow notation, have expressions that are evaluated from the right.
 
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: \(a \uparrow_{\uparrow_2} b = a \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_b} a = a \uparrow_{b+1} b\)
 +
 +
Then: \(a \uparrow_{\uparrow_{\uparrow_2}} b = a \uparrow_{\uparrow_{b+1}} b\) 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 22:25, 3 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: \(a \uparrow_{\uparrow_2} b = a \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_b} a = a \uparrow_{b+1} b\)

Then: \(a \uparrow_{\uparrow_{\uparrow_2}} b = a \uparrow_{\uparrow_{b+1}} b\) 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