User blog:Googleaarex/Extended 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.

Dimensional Array Up-arrow Notation
Next, \(a \uparrow_{,_2 \uparrow\#} b\) = \(a \uparrow_{,..., \uparrow ,_2 \#} a\) (a ,'s)

The def. continue in next part.

Nested Array Up-arrow Notation
More coming soon?