FANDOM


Here is all extensions of Up-Arrow Notation.

Extended Up-arrow Notation

We defined \(a \uparrow_2 b\), which is equal to \(a \uparrow ... \uparrow a\) (b \(\uparrow\)'s).

Then \(a \uparrow\uparrow_2 b\) = \(a \uparrow_2 a ... a \uparrow_2 a\) (b a's), \(a \uparrow\uparrow\uparrow_2 b\) = \(a \uparrow\uparrow_2 a ... a \uparrow\uparrow_2 a\) (b a's), etc.

Next, \(a \uparrow_2\uparrow_2 b\) = \(a \uparrow ... \uparrow\uparrow_2 a\) (b \(\uparrow\)'s), \(a \uparrow_2\uparrow_2\uparrow_2 b\) = \(a \uparrow ... \uparrow\uparrow_2\uparrow_2 a\) (b \(\uparrow\)'s), etc.

The next arrow type is \(\uparrow_3\). \(a \uparrow_3 b\) = \(a \uparrow_2 ... \uparrow_2 a\) (b \(\uparrow_2\)'s). Then \(\uparrow_4\), \(\uparrow_5\), etc.

Rules

1. \(a \uparrow b\) = \(a^b\)

2. \(a \uparrow \# b\) = \(a \# (a \uparrow \# b-1)\)

3. \(a \uparrow_1 \# b\) = \(a \uparrow \# b\)

4. \(a \uparrow_c \# b\) = \(a \uparrow_{c-1} ... \uparrow_{c-1} \# a\) (b \(\uparrow_{c-1}\)'s)

Levels

\(\uparrow_2\) has level \(\omega\)

\(\uparrow^n\uparrow_2\) has level \(\omega + n\)

\(\uparrow_2\uparrow_2\) has level \(\omega 2\)

\(\uparrow_2^n\) has level \(\omega n\)

\(\uparrow_3\) has level \(\omega^2\)

\(\uparrow_3^n\) has level \(\omega^2 n\)

\(\uparrow_n\) has level \(\omega^{n-1}\)

Limit is level \(\omega^\omega\)

Nested Up-arrow Notation

Next, \(\uparrow_2\) will be \(\uparrow_{\uparrow\uparrow}\), \(\uparrow_3\) will be \(\uparrow_{\uparrow\uparrow\uparrow}\), etc.

Then, \(a \uparrow_{\uparrow_2} b\) = \(a \uparrow_{\uparrow ... \uparrow} a\) (b \(\uparrow\)'s), \(a \uparrow_{\uparrow\uparrow_2} b\) = \(a \uparrow_{\uparrow_2} ... \uparrow_{\uparrow_2} a\) (b \(\uparrow_{\uparrow_2}\)'s), etc.

Also, \(a \uparrow_{\uparrow_c} b\) = \(a \uparrow_{\uparrow_{c-1} ... \uparrow_{c-1}} a\) (b \(\uparrow_{c-1}\)'s) and \(a \uparrow_{\uparrow_{\uparrow_2}} b\) = \(a \uparrow_{\uparrow_{\uparrow ... \uparrow}} a\) (b \(\uparrow\)'s).

Finally, \(\uparrow_{\uparrow_{\uparrow_{\uparrow_\#}}}\) is 4 levels, \(\uparrow_{\uparrow_{\uparrow_{\uparrow_{\uparrow_\#}}}}\) is 5 levels, etc.

Rules

3. \(a \#_{\uparrow_\uparrow} \# b\) = \(a \#_{\uparrow} \# b\)

4. \(a \#_{\uparrow_{\uparrow\#}} \# b\) = \(a \#_{\uparrow_\# ... \uparrow_\# \#} a\) (b \(\uparrow_{c-1}\)'s)

Levels

\(\uparrow_{\uparrow_2}\) has level \(\omega^\omega\)

\(\uparrow_{\uparrow^n\uparrow_2}\) has level \(\omega^{\omega + n}\)

\(\uparrow_{\uparrow_2^n}\) has level \(\omega^{\omega n}\)

\(\uparrow_{\uparrow_n}\) has level \(\omega^{\omega^{n-1}}\)

\(\uparrow_{\uparrow_{\uparrow_2}}\) has level \(\omega^{\omega^\omega}\)

\(\uparrow_{..._{\uparrow_2}}\) (n \(\uparrow\)'s) has level \(^n\omega\)

Limit is level \(\varepsilon_0\)

Array Up-arrow Notation

We have \(a \uparrow_{,\uparrow} b\), which is equal to \(a \uparrow_{..._{\uparrow_b}} a\) (b levels).

Then \(a \uparrow_{\uparrow,\uparrow} b\) = \(a \uparrow_{,\uparrow} ... \uparrow_{,\uparrow} a\) (b \(\uparrow_{,\uparrow}\)'s), \(a \uparrow_{\uparrow\uparrow,\uparrow} b\) = \(a \uparrow_{\uparrow,\uparrow} ... \uparrow_{\uparrow,\uparrow} a\) (b \(\uparrow_{\uparrow,\uparrow}\)'s), etc. And \(a \uparrow_{,\#\uparrow} b\), which is equal to \(a \uparrow_{..._{\uparrow_{b,\#}}...,\#} a\) (b levels).

There is 3 entries, 4 entries, etc. So \(a \uparrow_{\circ,\#\uparrow} b\), which is equal to \(a \uparrow_{\circ ..._{\circ \uparrow_{\circ b,\#}}...,\#} a\) (b levels).

\(\circ\) is a row of commas.

Levels

\(\uparrow_{,\uparrow}\) has level \(\varepsilon_0\)

\(\uparrow_{,\uparrow}\uparrow_{,\uparrow}\) has level \(\varepsilon_0 2\)

\(\uparrow_{\uparrow,\uparrow}\) has level \(\varepsilon_0 \omega\)

\(\uparrow_{\uparrow_{,\uparrow},\uparrow}\) has level \(\varepsilon_0^2\)

\(\uparrow_{\uparrow_{\uparrow_{,\uparrow},\uparrow},\uparrow}\) has level \(\varepsilon_0^{\varepsilon_0}\)

\(\uparrow_{,\uparrow\uparrow}\) has level \(\varepsilon_1\)

\(\uparrow_{,\uparrow^n}\) has level \(\varepsilon_{n-1}\)

\(\uparrow_{,\uparrow_{\uparrow\uparrow}}\) has level \(\varepsilon_\omega\)

\(\uparrow_{,\uparrow_{,\uparrow}}\) has level \(\varepsilon_{\varepsilon_0}\)

\(\uparrow_{,,\uparrow}\) has level \(\zeta_0\)

\(\uparrow_{,,\uparrow^n}\) has level \(\zeta_{n-1}\)

\(\uparrow_{,,\uparrow_{,,\uparrow}}\) has level \(\zeta_{\zeta_0}\)

\(\uparrow_{,,,\uparrow}\) has level \(\eta_0\)

\(\uparrow_{,...,\uparrow}\) (n ,'s) has level \(\varphi(n,0)\)

Limit is level \(\varphi(\omega,0)\)

Dimensional Array Up-arrow Notation

Next, \(a \uparrow_{,_2 \uparrow\#} b\) = \(a \uparrow_{,..., \uparrow ,_2 \#} a\) (b ,'s)

Then, \(a \uparrow_{,_2,_2 \uparrow\#} b\) = \(a \uparrow_{,_2,..., \uparrow ,_2 \#} a\) (b ,'s), \(a \uparrow_{,_3 \uparrow\#} b\) = \(a \uparrow_{,_2...,_2 \uparrow ,_3 \#} a\) (b ,'s), etc.

If \(,_\#\) don't have \(\uparrow_\uparrow\)'s, \(a \uparrow_{,_{\uparrow_{\uparrow\#}} \uparrow\#} b\) = \(a \uparrow_{,_{\uparrow_\# ... \uparrow_\#} \uparrow ,_{\uparrow_{\uparrow\#}} \#} a\) (b \(\uparrow_\#\)'s)

The def. continue in next part.

Levels

\(\uparrow_{,_2\uparrow}\) has level \(\varphi(\omega,0)\)

\(\uparrow_{,_2,\uparrow}\) has level \(\varphi(\omega + 1,0)\)

\(\uparrow_{,_2,...,\uparrow}\) (n ,'s) has level \(\varphi(\omega + n,0)\)

\(\uparrow_{,_2,_2\uparrow}\) has level \(\varphi(\omega 2,0)\)

\(\uparrow_{,_2...,_2\uparrow}\) (n \(,_2\)'s) has level \(\varphi(\omega n,0)\)

\(\uparrow_{,_3\uparrow}\) has level \(\varphi(\omega^2,0)\)

\(\uparrow_{,_n\uparrow}\) has level \(\varphi(\omega^{n-1},0)\)

\(\uparrow_{,_{\uparrow_2}\uparrow}\) has level \(\varphi(\omega^\omega,0)\)

\(\uparrow_{,_{\uparrow_{,\uparrow}}\uparrow}\) has level \(\varphi(\varepsilon_0,0)\)

\(\uparrow_{,_{\uparrow_{,_2\uparrow}}\uparrow}\) has level \(\varphi(\varphi(\omega,0),0)\)

\(\uparrow_{,_{\uparrow_{,_{\uparrow_{,_2\uparrow}}\uparrow}}\uparrow}\) has level \(\varphi(\varphi(\varphi(\omega,0),0),0)\)

Limit is level \(\Gamma_0\), thanks Wythagoras!

Nested Array Up-arrow Notation

Also, \(a \uparrow_{,_{,\uparrow} \#} b\) = \(a \uparrow_{,_{...\uparrow_{,_b \#}...} \#} a\) (b levels), \(a \uparrow_{,_{,_2 \uparrow} \#} b\) = \(a \uparrow_{,_{,...,\uparrow} \#} b\) (b ,'s), etc. This is 2 nested levels.

We have 3 nested levels (\(a \uparrow_{,_{,_{,\uparrow} \#} \#} b\) = \(a \uparrow_{,_{,_{...\uparrow_{,_{,_b} \# \#}...} \#} \#} a\) (b levels)), 4 nested levels, etc.

Levels

\(\uparrow_{,_{,\uparrow}\uparrow}\) has level \(\Gamma_0\)

\(\uparrow_{,_{,\uparrow},\uparrow}\) has level \(\varphi(1,1,0)\)

\(\uparrow_{,_{,\uparrow},_2\uparrow}\) has level \(\varphi(1,\omega,0)\)

\(\uparrow_{,_{,\uparrow},_{,_{,\uparrow}\uparrow}\uparrow}\) has level \(\varphi(1,\Gamma_0,0)\)

\(\uparrow_{,_{,\uparrow},_{,\uparrow}\uparrow}\) has level \(\varphi(2,0,0)\)

\(\uparrow_{,_{,\uparrow}...,_{,\uparrow}\uparrow}\) (n \(,_{,\uparrow}\)'s) has level \(\varphi(n,0,0)\)

\(\uparrow_{,_{\uparrow,\uparrow}\uparrow}\) has level \(\varphi(\omega,0,0)\)

\(\uparrow_{,_{\uparrow^n,\uparrow}\uparrow}\) has level \(\varphi(\omega^n,0,0)\)

\(\uparrow_{,_{\uparrow_{,\uparrow},\uparrow}\uparrow}\) has level \(\varphi(\varepsilon_0,0,0)\)

\(\uparrow_{,_{\uparrow_{,_{,\uparrow}\uparrow},\uparrow}\uparrow}\) has level \(\varphi(\Gamma_0,0,0)\)

\(\uparrow_{,_{,\uparrow\uparrow}\uparrow}\) has level \(\varphi(1,0,0,0)\)

\(\uparrow_{,_{,\uparrow^n}\uparrow}\) has level \(\vartheta(\Omega^n)\)

\(\uparrow_{,_{,\uparrow_2}\uparrow}\) has level \(\vartheta(\Omega^\omega)\)

\(\uparrow_{,_{,\uparrow_{,\uparrow}}\uparrow}\) has level \(\vartheta(\Omega^{\varepsilon_0})\)

\(\uparrow_{,_{,\uparrow_{,_{,\uparrow^n}\uparrow}}\uparrow}\) has level \(\vartheta(\Omega^{\vartheta(\Omega^n)})\)

\(\uparrow_{,_{,\uparrow_{,_{,\uparrow_2}\uparrow}}\uparrow}\) has level \(\vartheta(\Omega^{\vartheta(\Omega^\omega)})\)

\(\uparrow_{,_{,,\uparrow}\uparrow}\) has level \(\vartheta(\Omega^\Omega)\)

\(\uparrow_{,_{,\uparrow,\uparrow}\uparrow}\) has level \(\vartheta(\Omega^{\Omega + 1})\)

\(\uparrow_{,_{,,\uparrow\uparrow}\uparrow}\) has level \(\vartheta(\Omega^{\Omega 2})\)

\(\uparrow_{,_{,,,\uparrow}\uparrow}\) has level \(\vartheta(\Omega^{\Omega^2})\)

\(\uparrow_{,_{,...,\uparrow}\uparrow}\) (n ,'s) has level \(\vartheta(\Omega^{\Omega^{n-1}})\)

\(\uparrow_{,_{,_2\uparrow}\uparrow}\) has level \(\vartheta(\Omega^{\Omega^\omega})\)

\(\uparrow_{,_{,_{,\uparrow}\uparrow}\uparrow}\) has level \(\vartheta(\Omega^{\Omega^\Omega})\)

\(\uparrow_{,_{,_{,...,\uparrow}\uparrow}\uparrow}\) (n ,'s) has level \(\vartheta(\Omega^{\Omega^{\Omega^n}})\)

\(\uparrow_{,_{,_{,_{,\uparrow}\uparrow}\uparrow}\uparrow}\) has level \(\vartheta(\Omega^{\Omega^{\Omega^\Omega}})\)

Limit is level \(\vartheta(\varepsilon_{\Omega + 1})\)

Seperators Array Up-arrow Notation

The first seperator is \(,\). And the second seperator is \([\uparrow\uparrow]\).

So, \(a \uparrow_{[\uparrow\uparrow]\#\uparrow} b\) = \(a \uparrow_{,_{... ,_b\uparrow[\uparrow\uparrow]\# ...}\uparrow[\uparrow\uparrow]\#} a\) (b nested levels).

Then \(a \uparrow_{[\uparrow\uparrow][\uparrow\uparrow]\#\uparrow} b\) = \(a \uparrow_{[\uparrow\uparrow],_{... [\uparrow\uparrow],_b\uparrow[\uparrow\uparrow]\# ...}\uparrow[\uparrow\uparrow]\#} a\) (b nested levels), \(a \uparrow_{[\uparrow\uparrow][\uparrow\uparrow][\uparrow\uparrow]\#\uparrow} b\) = \(a \uparrow_{[\uparrow\uparrow][\uparrow\uparrow],_{... [\uparrow\uparrow][\uparrow\uparrow],_b\uparrow[\uparrow\uparrow]\# ...}\uparrow[\uparrow\uparrow]\#} a\) (b nested levels), etc.

Sometimes there dimensions of 2nd seperator, \([\uparrow\uparrow]_\#\).

Next, there is 3rd seperator. \(a \uparrow_{[\uparrow\uparrow\uparrow]\#\uparrow} b\) = \(a \uparrow_{[\uparrow\uparrow]_{... [\uparrow\uparrow]_b\uparrow[\uparrow\uparrow\uparrow]\# ...}\uparrow[\uparrow\uparrow\uparrow]\#} a\) (b nested levels).

In general, \(a \uparrow_{[\uparrow\#]\#\uparrow} b\) = \(a \uparrow_{[\#]_{... [\#]_b\uparrow[\uparrow\#]\# ...}\uparrow[\uparrow\#]\#} a\) (b nested levels).

The limit is \(\uparrow_\alpha\), \(\alpha \rightarrow [\alpha]\uparrow\)

Levels

\(\uparrow_{[\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega + 1})\)

\(\uparrow_{,\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\varepsilon_{\vartheta(\varepsilon_{\Omega + 1})+1}\)

\(\uparrow_{,_2\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\varphi(\omega,\vartheta(\varepsilon_{\Omega + 1})+1)\)

\(\uparrow_{,_n\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\varphi(\omega^{n-1},\vartheta(\varepsilon_{\Omega + 1})+1)\)

\(\uparrow_{,_{\uparrow_2}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\varphi(\omega^\omega,\vartheta(\varepsilon_{\Omega + 1})+1)\)

\(\uparrow_{,_{\uparrow_{,\uparrow}}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\varphi(\varepsilon_0,\vartheta(\varepsilon_{\Omega + 1})+1)\)

\(\uparrow_{,_{\uparrow_{[\uparrow\uparrow]\uparrow}}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\varphi(\vartheta(\varepsilon_{\Omega + 1}),1)\)

\(\uparrow_{,_{,\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\Gamma_{\vartheta(\varepsilon_{\Omega + 1})+1}\)

\(\uparrow_{,_{,\uparrow\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\varphi(1,0,0,\vartheta(\varepsilon_{\Omega + 1})+1)\)

\(\uparrow_{,_{,\uparrow_2}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\Omega^\omega,\vartheta(\varepsilon_{\Omega + 1})+1)\)

\(\uparrow_{,_{,,\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\Omega^\Omega,\vartheta(\varepsilon_{\Omega + 1})+1)\)

\(\uparrow_{,_{[\uparrow\uparrow]\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega + 1},1)\)

\(\uparrow_{,_{[\uparrow\uparrow]\uparrow},_{[\uparrow\uparrow]\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega + 1}2)\)

\(\uparrow_{,_{,\uparrow[\uparrow\uparrow]\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega + 1}\Omega)\)

\(\uparrow_{,_{,_{[\uparrow\uparrow]\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega + 1}^2)\)

\(\uparrow_{[\uparrow\uparrow]\uparrow\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega + 2})\)

\(\uparrow_{[\uparrow\uparrow]\uparrow_2}\) has level \(\vartheta(\varepsilon_{\Omega + \omega})\)

\(\uparrow_{[\uparrow\uparrow],\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega 2})\)

\(\uparrow_{[\uparrow\uparrow],_{[\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\varepsilon_{\varepsilon_{\Omega + 1}})\)

\(\uparrow_{[\uparrow\uparrow][\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\zeta_{\Omega + 1})\)

\(\uparrow_{[\uparrow\uparrow]_2\uparrow}\) has level \(\vartheta(\vartheta_1(\omega))\)

\(\uparrow_{[\uparrow\uparrow]_n\uparrow}\) has level \(\vartheta(\vartheta_1(\omega^{n-1}))\)

\(\uparrow_{[\uparrow\uparrow]_{,\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega))\)

\(\uparrow_{[\uparrow\uparrow]_{,_{[\uparrow\uparrow]\uparrow}\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\varepsilon_{\Omega + 1}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow}[\uparrow\uparrow]_{,_{[\uparrow\uparrow]\uparrow}\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2 + \vartheta_1(\Omega_2)))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow}[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2 2))\)

\(\uparrow_{[\uparrow\uparrow]_{,\uparrow[\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2 \Omega))\)

\(\uparrow_{[\uparrow\uparrow]_{,_{[\uparrow\uparrow]\uparrow}\uparrow[\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2 \varepsilon_{\Omega + 1}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^2))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow^n}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^n))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow],\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^\Omega))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow][\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow][\uparrow\uparrow]\uparrow}[\uparrow\uparrow]_{[\uparrow\uparrow]}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2}+\Omega_2))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow][\uparrow\uparrow]\uparrow}[\uparrow\uparrow]_{[\uparrow\uparrow][\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2} 2))\)

\(\uparrow_{[\uparrow\uparrow]_{\uparrow^n[\uparrow\uparrow][\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2} \omega^n))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow^n[\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2 + n}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow][\uparrow\uparrow]\uparrow^n}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2 n}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow][\uparrow\uparrow][\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2^2}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]_{2}\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2^\omega}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]_{,\uparrow}\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2^\Omega}))\)

\(\uparrow_{[\uparrow\uparrow]_{[\uparrow\uparrow]_{[\uparrow\uparrow]\uparrow}\uparrow}\uparrow}\) has level \(\vartheta(\vartheta_1(\Omega_2^{\Omega_2^{\Omega_2}}))\)

\(\uparrow_{[\uparrow\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega_2 + 1})\)

\(\uparrow_{[\uparrow\uparrow\uparrow]\uparrow_2}\) has level \(\vartheta(\varepsilon_{\Omega_2 + \omega})\)

\(\uparrow_{[\uparrow\uparrow\uparrow],\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega_2 + \Omega})\)

\(\uparrow_{[\uparrow\uparrow\uparrow][\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega_2 2})\)

\(\uparrow_{[\uparrow\uparrow\uparrow][\uparrow\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\zeta_{\Omega_2 + 1})\)

\(\uparrow_{[\uparrow\uparrow\uparrow]_2\uparrow}\) has level \(\vartheta(\vartheta_2(\omega))\)

\(\uparrow_{[\uparrow\uparrow\uparrow]_{[\uparrow\uparrow\uparrow]\uparrow}\uparrow}\) has level \(\vartheta(\Omega_3)\)

\(\uparrow_{[\uparrow\uparrow\uparrow\uparrow]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega_3 + 1})\)

\(\uparrow_{[\uparrow^n]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega_{n-1} + 1})\)

\(\uparrow_{[\uparrow_2]\uparrow}\) has level \(\vartheta(\Omega_\omega)\)

\(\uparrow_{[\uparrow\uparrow_2]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega_\omega + 1})\)

\(\uparrow_{[\uparrow^n\uparrow_2]\uparrow}\) has level \(\vartheta(\varepsilon_{\Omega_{\omega + (n-1)} + 1})\)

\(\uparrow_{[\uparrow_2^n]\uparrow}\) has level \(\vartheta(\Omega_{\omega n})\)

\(\uparrow_{[\uparrow_n]\uparrow}\) has level \(\vartheta(\Omega_{\omega^{n-1}})\)

\(\uparrow_{[,\uparrow]\uparrow}\) has level \(\vartheta(\Omega_\Omega)\)

\(\uparrow_{[,_{[\uparrow\uparrow]\uparrow}\uparrow]\uparrow}\) has level \(\vartheta(\Omega_{\varepsilon_{\Omega + 1}})\)

\(\uparrow_{[[\uparrow\uparrow]\uparrow]\uparrow}\) has level \(\vartheta(\Omega_{\Omega_2})\)

\(\uparrow_{[[\uparrow^n]\uparrow]\uparrow}\) has level \(\vartheta(\Omega_{\Omega_n})\)

\(\uparrow_{[[,\uparrow]\uparrow]\uparrow}\) has level \(\vartheta(\Omega_{\Omega_\Omega})\)

\(\uparrow_{[[[\uparrow^n]\uparrow]\uparrow]\uparrow}\) has level \(\vartheta(\Omega_{\Omega_{\Omega_n}})\)

Limit is level \(\psi(\psi_I(0))\)

Hyperarray Up-arrow Notation

We have 2-hyperentries seperator. \(a \uparrow_{[\&\uparrow]\uparrow\#} b\) = \(a \uparrow_{[... \uparrow[\&\uparrow]\# ...]\uparrow[\&\uparrow]\#} a\) (b nested), \(a \uparrow_{[\&\uparrow\uparrow]\uparrow\#} b\) = \(a \uparrow_{[[\&\uparrow]...[\&\uparrow]\uparrow[\&\uparrow\uparrow]\# ...]\uparrow[\&\uparrow\uparrow]\#} a\) (b nested), etc.

Also we have 3-hyperentries seperator. \(a \uparrow_{[\&\&\uparrow]\uparrow\#} b\) = \(a \uparrow_{[\& ... [\&\uparrow[\&\&\uparrow]\#]\uparrow[\&\&\uparrow]\# ...]\uparrow[\&\&\uparrow]\#} b\) (n nested), \(a \uparrow_{[\&\&\uparrow\uparrow]\uparrow\#} b\) = \(a \uparrow_{[\& ... [\&\uparrow[\&\&\uparrow\uparrow]\#\&\uparrow]\uparrow[\&\&\uparrow\uparrow]\# \&\uparrow...]\uparrow[\&\&\uparrow]\#\&\uparrow} b\) (n nested), etc.

Then an 4-hyperentries seperator, 5-hyperentries seperator, etc.

We have dimensional hyperentries and hyperseperators.

Seperators change \([\alpha]\) into \(,[\alpha]\).

Limit is \(\uparrow_{,[\alpha]\uparrow}, \alpha \rightarrow \&[\alpha]\uparrow\)

Levels

\(\uparrow_{,[\&\uparrow]\uparrow}\) has level \(\psi(\psi_I(0))\)

\(\uparrow_{,[\&\uparrow],[\&\uparrow]\uparrow}\) has level \(\psi(I)\)

\(\uparrow_{,[\&\uparrow]_2\uparrow}\) has level \(\psi(I^\omega)\)

\(\uparrow_{,[\&\uparrow]_{,[\&\uparrow]\uparrow}\uparrow}\) has level \(\psi(I^I)\)

\(\uparrow_{,[\uparrow\&\uparrow]\uparrow}\) has level \(\psi(\varepsilon_{I+1})\)

\(\uparrow_{,[\uparrow_2\&\uparrow]\uparrow}\) has level \(\psi(\Omega_{I+\omega})\)

\(\uparrow_{,[,\uparrow\&\uparrow]\uparrow}\) has level \(\psi(\Omega_{I+\Omega})\)

\(\uparrow_{,[,[\&\uparrow]\uparrow\&\uparrow]\uparrow}\) has level \(\psi(\Omega_{I+\psi_I(0)})\)

\(\uparrow_{,[,[,[\&\uparrow]\uparrow\&\uparrow]\uparrow\&\uparrow]\uparrow}\) has level \(\psi(\Omega_{I+\psi_{\Omega_{I+\psi_I(0)}}(0)})\)

\(\uparrow_{,[\&\uparrow\uparrow]\uparrow}\) has level \(\psi(\Omega_{I2})\)

\(\uparrow_{,[\&\&\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^2})\)

\(\uparrow_{,[\&_{\uparrow\uparrow}\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^\omega})\)

\(\uparrow_{,[\&_{,\uparrow}\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^\Omega})\)

\(\uparrow_{,[\&_{\&\uparrow}\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^I})\)

\(\uparrow_{,[\&_{\uparrow\&\uparrow}\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^{I\omega}})\)

\(\uparrow_{,[\&_{[\&\uparrow]\uparrow\&\uparrow}\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^{I\psi_I(0)}})\)

\(\uparrow_{,[\&_{\&\uparrow_2}\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^{I^\omega}})\)

\(\uparrow_{,[\&_{\&\&\uparrow}\uparrow]\uparrow}\) has level \(\psi(\Omega_{I^{I^I}})\)

\(\uparrow_{,[\&[\uparrow\uparrow]\uparrow]\uparrow}\) has level \(\psi(\Omega_{\varepsilon_{I+1}})\)

\(\uparrow_{,[\&[\uparrow\uparrow\uparrow]\uparrow]\uparrow}\) has level \(\psi(\Omega_{\varepsilon_{\Omega_{I+1}+1}})\)

\(\uparrow_{,[\&[\&\uparrow]\uparrow]\uparrow}\) has level \(\psi(\Omega_{I2})\)

\(\uparrow_{,[\&[\&[\&\uparrow]\uparrow]\uparrow]\uparrow}\) has level \(\psi(\Omega_{\Omega_{I2}})\)

Limit is level \(\psi(\psi_{I_2}(0))\)

Multi-type Entry Up-arrow Notation

1-Entry is Entry, 2-Entry is Hyperentry, 3-Entry is Hyperhyperentry, etc.

So \(<1>\) = \(,\) and \(<2>\) = \(\&\).

We defined \(a \uparrow_{[[/\uparrow]\&\uparrow],\uparrow} b\), which is equal to \(a \uparrow_{[[... [\&\uparrow] ...]\&\uparrow],\uparrow} a\) (b nested). Then \(a \uparrow_{[\text{[}//\uparrow]\&\uparrow],\uparrow} b\) = \(a \uparrow_{[\text{[}/... [/\&\uparrow] ...]\&\uparrow],\uparrow} a\) (b nested), \(a \uparrow_{[\text{[}///\uparrow]\&\uparrow],\uparrow} b\) = \(a \uparrow_{[\text{[}//... \text{[}//\&\uparrow] ...]\&\uparrow],\uparrow} a\) (b nested), etc.

Then we have dimensional hyperentries, hyperhyperseperators, hyperhyperentries, dimensional hyperhyperentries, etc.

So \(a \uparrow_{[[[\neg\uparrow]/\uparrow]\&\uparrow],\uparrow} b\) = \(a \uparrow_{???} b\).

It have too many symbols, so we shortened this notation. \(/ = <3>\) and \(\neg = <4>\).

New rule: \(\uparrow_{\#<x>\# ... \#<x>\#}\) = \(\uparrow_{[\#<x>\# ... \#<x>\#]<x-1>\uparrow}\)

We have array-entry (\(<,\uparrow>\)), dimensional-entry (\(<,_2\uparrow>\)), seperator-entry (\(<[\uparrow\uparrow],\uparrow>\)), hyperentry-entry (\(<[\&\uparrow],\uparrow>\)), hyperhyperentry-entry, etc.

Levels

\(\uparrow_{,[<2>[<3>\uparrow]\uparrow]\uparrow}\) has level \(\psi(\psi_{I_2}(0))\)

\(\uparrow_{,[<2>[<3>[<4>\uparrow]\uparrow]\uparrow]\uparrow}\) has level \(\psi(\psi_{I_3}(0))\)

\(\uparrow_{<n>\uparrow}\) has level \(\psi(\psi_{I_{n-1}}(0))\)

\(\uparrow_{<\uparrow_{\uparrow\uparrow}>\uparrow}\) has level \(\psi(\psi_{I_\omega}(0))\)

\(\uparrow_{<,\uparrow>\uparrow}\) has level \(\psi(\psi_{I_\Omega}(0))\)

\(\uparrow_{<<2>\uparrow>\uparrow}\) has level \(\psi(\psi_{I_I}(0))\)

\(\uparrow_{<<n>\uparrow>\uparrow}\) has level \(\psi(\psi_{I_{I_{n-1}}(0))\)

\(\uparrow_{<<,\uparrow>\uparrow>\uparrow}\) has level \(\psi(\psi_{I_{I_\Omega}}(0))\)

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

Angle Array Up-arrow Notation

We defined \(a \uparrow_{<><\uparrow>\uparrow} b\), which is equal to \(a \uparrow_{<<...<<>\uparrow>...>\uparrow>\uparrow} a\) (b nested). Then \(a \uparrow_{<><\uparrow>\uparrow\uparrow} b\) = \(a \uparrow_{<... \uparrow<\><\uparrow>\uparrow ...>\uparrow<><\uparrow>\uparrow} a\) (b nested) and any arrows work same.

So there is new rule: \(\# <0> \#\) = \(\# \#\).

So \(a \uparrow_{<><\uparrow\uparrow>\uparrow} b\) = \(a \uparrow_{<... <><\uparrow>\uparrow ...><\uparrow>\uparrow} a\) (b nested)

Preview:

\(<><\#\uparrow>\) = \(<<...<<\uparrow><\#>><\#>...><\#>><\#>\)

\(<><><\#\uparrow>\) = \(<><<><...<><<><\uparrow><\#>><\#>...><\#>><\#>\)

\(\#<><\#\uparrow>\) = \(\#<\#<...\#<\#<\uparrow><\#>><\#>...><\#>><\#>\)

\(<\#\uparrow>_{<\uparrow\uparrow>}\) = \(<>...<><\uparrow><\#>_{<\uparrow\uparrow>}\)

\(<>_{<\uparrow\uparrow>}<\#\uparrow>\) = \(<... <\uparrow>_{<\uparrow\uparrow>}<\#> ...>_{<\uparrow\uparrow>}<\#>\)

\(<>_{<\uparrow\uparrow>}<\#\uparrow>_{<\uparrow\uparrow>}\) = \(<>_{<\uparrow\uparrow>}<>...<><\uparrow><\#>_{<\uparrow\uparrow>}\)

\(<\#\uparrow>_{<\uparrow\uparrow\uparrow>}\) = \(<>_{<\uparrow\uparrow>}...<>_{<\uparrow\uparrow>}<\uparrow>_{<\uparrow\uparrow>}<\#>_{<\uparrow\uparrow\uparrow>}\)

\(<\#\uparrow>_{<\uparrow\#>}\) = \(<>_{<\#>}...<>_{<\#>}<\uparrow>_{<\#>}<\#>_{<\uparrow\#>}\)

\(<\uparrow>_{<><\uparrow>}\) = \(<\uparrow>_{<...<\uparrow>_{<\uparrow>}...>}\)

We have nested <>'s.

Legion Array Up-arrow Notation

Preview: \(<>,<\uparrow>\) = \(<\uparrow>_{...<\uparrow>...}\)

\(<>,<\#\uparrow>\) = \(<\uparrow>_{...<\uparrow><\#>...}<\#>\)

\(<>,<>,<\#\uparrow>\) = \(<>,<\uparrow>_{...<>,<\uparrow><\#>...}<\#>\)

\(<>,_2<\#\uparrow>\) = \(<>,<>...<>,<>,<\uparrow>,_2<\#\uparrow>\)

\(<>[\uparrow\uparrow]<\uparrow>\) = \(<>,_{... <>,<\uparrow> ...}<\uparrow>\)

\(<>\&<\uparrow>\) = \(<>,[...<>,<\uparrow>...]<\uparrow>\)

\(<>(<3>)<\uparrow>\) = \(<>\&[...<>\&<\uparrow>...]<\uparrow>\)

\(<>(<\uparrow_2>)<\uparrow>\) = \(<>(<\uparrow...\uparrow>)<\uparrow>\)

\(<>(<><\uparrow>)<\uparrow>\) = \(<>(...<>(<>\uparrow)<\uparrow>...\uparrow)<\uparrow>\)

\(<>(<>,<\uparrow>)<\uparrow>\) = \(<>(<><\uparrow>_{...<>(<><\uparrow>)<\uparrow>...})<\uparrow>\)

More Coming Soon?

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