Extended arrow notation

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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

\(\Omega\) typed arrows

Limit: \(\psi(\varepsilon_{\Omega+1})\)

\(\Omega_2\) typed arrows

Limit: \(\psi(\psi_1(\varepsilon_{\Omega_2+1}))\)

\(\Omega_3\) typed arrows and beyond

Limit: \(\psi(\psi_I(0))\)

Inaccesible arrows

Limit: \(\psi(\psi_{I(1,0)}(0))\)

1-inaccesible arrows and beyond

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

Dimensional array up-arrow notation

Limit: \(\psi(\psi_{\chi(\varepsilon_{M+1})}(0))\)

Hyperarray up-arrow notation

Limit: \(\psi(\psi_{\chi(M(1,0))}(0))\)

Legion array up-arrow notation

Layered arrays

Limit: \(\psi(\psi_{{\Xi(1)}^\omega}(0))\)

The hyperseparator

Limit: \(\psi(\psi_{M(1,\Xi(1)+1)}(0))\)

The second hyperseparator


Hyperlegion array up-arrow notation