## FANDOM

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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?