Difference between revisions of "Extended arrow notation"
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Latest revision as of 21:05, 19 July 2021
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.
\(a \uparrow_2 b = a \underbrace{\uparrow\uparrow\dots\uparrow}_b a\)
\(a \uparrow_3 b = a \underbrace{\uparrow_2\uparrow_2\dots\uparrow_2}_b a\)
\(a \uparrow_n b = a \underbrace{\uparrow_{n1}\uparrow_{n1}\dots\uparrow_{n1}}_b a\)
Note that all parts of Extended arrow notation, like Knuth's uparrow notation, have expressions that are evaluated from the right.
Limit in FGH: \(f_\omega(n)\)
Nested uparrow notation
To extend the notation here, we first have to make a change: \(\uparrow_n = \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_{n1}}\)
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 uparrow 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: \(\psi(\psi_{I(1,0)}(0))\)
1inaccesible arrows and beyond
Limit: \(\psi(\psi_{I(\omega, 0)}(0))\)
Dimensional array uparrow notation
Limit: \(\psi(\psi_{\chi(\varepsilon_{M+1})}(0))\)
Hyperarray uparrow notation
Limit: \(\psi(\psi_{\chi(M(1,0))}(0))\)
Legion array uparrow notation
Layered arrays
Limit: \(\psi(\psi_{{\Xi(1)}^\omega}(0))\)
The hyperseparator
Limit: \(\psi(\psi_{M(1,\Xi(1)+1)}(0))\)
The second hyperseparator
Limit:???