User:Wythagoras/Dollar function/Extended Bracket notation

This is the extended version of Bracket Notation, and the second part of Dollar Function. It is similiar to \(\Omega\)s in FGH and the extended Buchholz Hydra.

Formal definition
\(\bullet\) can be anything

\(\circ\) is a group of brackets. 1. If there is nothing after the $, the array is solved. The value of the array is the number before the $.

2. \(a\$b\bullet=(a+b)\$\bullet\)

3. \(a\$\circ[0]\bullet\circ=a\$\circ a\bullet\circ\)

4. \(a\$\circ[\bullet+1]_c\bullet\circ=a\$\circ[\bullet]_c[\bullet]_c...[\bullet]_c[\bullet]_c\bullet\circ\) with a \(\bullet\)'s

5. If the bracket contains a zero and the bracket has other content, you can remove the zero.

6. If the active bracket has level k and a zero in it, search for the least nested bracket with level (k-1) with the same array in it, nest that bracket a times in the place of the level k bracket and put [0] on the place where the active bracket was.

S1. The outermost bracket is always level 1

S2. If there is no bracket with level (k-1), add it directly after the level k bracket.

The active bracket is the bracket with the lowest level. The brackets can be ordered by level in FGH, and then removing outermost bracket. Or you look to: smallest bracket type, least number of nestings, smallest number inside the bracket.

Examples
\(a\$0]_2] = [[..[[0..]]\) a+1 nests

\(a\$0]_2[0 = [[0]_2a] = [[0]_2a-1][[0]_2a-1]...[[0]_2a-1][[0]_2a-1]\\) a brackets

\(a\$0]_2[[0]_2] = a\$0]_2[[..[[0..]]]\) a+1 nests

\(a\$0]_2[0]_2] = [[0]_2[[0]_2[[0]_2[...]]\) a nests

\(a\$[[1]_2] = a\$[[0]_2[0]_2...[0]_2[0]_2]\) a brackets

\(a\$[0]_2]_2 = a\$[[[...]_2]_2]]]_2]]\) a nests

\(a\$[[0]_3] = [[[0]_3]_2] = [[..[[0]_2]_2..]_2]_2]\) a nests

Analysis
\([[0]_2]\) has level \(\varepsilon_0\)

\([1[0]_2]\) has level \(\varepsilon_0\omega\)

\([[[0]_2][0]_2]\) has level \(\varepsilon_0^2\)

\([[1[0]_2][0]_2]\) has level \(\varepsilon_0^\omega\)

\([[0]_2[0]_2]\) has level \(\varepsilon_1\)

\([[1]_2]\) has level \(\varepsilon_\omega\)

\([0_2]\) has level \(\varepsilon_{\omega^2}\)

\([[0]_2_2]\) has level \(\varepsilon_{\varepsilon_0}\)

\([[[[0]_2_2]]_2]\) has level \(\varepsilon_{\varepsilon_{\varepsilon_0}}\)

\([[[0]_2]_2]\) has level \(\zeta_0\)

\([1[[0]_2]_2]\) has level \(\zeta_0\omega\)

\([0]_2]_2[1[[0]_2]_2\) has level \(\zeta_0^\omega\)

\([[[0]_2]_2[0]_2]\) has level \(\varepsilon_{\zeta_0+1}\)

\([[[0]_2]_2[1]_2]\) has level \(\varepsilon_{\zeta_0+\omega}\)

\([0]_2]_2[[[[0]_2]_2_2]\) has level \(\varepsilon_{\zeta_02}\)

\([[[0]_2]_2[[0]_2]_2]\) has level \(\zeta_1\)

\([[1[0]_2]_2]\) has level \(\zeta_\omega\)

\([[[[0]_2][0]_2]_2]\) has level \(\zeta_{\varepsilon_0}\)

\([[[[[0]_2]_2][0]_2]_2]\) has level \(\zeta_{\zeta_0}\)

\([[[0]_2[0]_2]_2]\) has level \(\eta_0\)

\([[[1]_2]_2]\) has level \(\varphi(\omega,0)\)

\( [[0_2]_2] \) has level \(\varphi(\omega^2,0)\)

\([[[0]_2_2]_2]\) has level \(\varphi(\varepsilon_0,0)\)

\([[[[0]_2]_2]_2]\) has level \(\vartheta(\Omega)\)

\([[1[[0]_2]_2]_2]\) has level \(\vartheta(\Omega+1)\)

\([[[0]_2[[0]_2]_2]_2]\) has level \(\vartheta(\Omega2)\)

\([[[1]_2[[0]_2]_2]_2]\) has level \(\vartheta(\Omega\omega)\)

\([[[[0]_2]_2[[0]_2]_2]_2]\) has level \(\vartheta(\Omega^2)\)

\([[[1[0]_2]_2]_2]\) has level \(\vartheta(\Omega^\omega)\)

\([[[[0]_2[0]_2]_2]_2]\) has level \(\vartheta(\Omega^\Omega)\)

\([[[[1]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega\omega})\)

\([[[[[0]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2})\)

\([[1[[[0]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2}+1)\)

\([[[[0]_2[[0]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2+\Omega})\)

\([[[[1]_2[[0]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2+\Omega\omega})\)

\([[[[1[0]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2\omega})\)

\([[[[[0]_2[0]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^3})\)

\([[[[[1]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^\omega})\)

\([[[[[[0]_2]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^\Omega})\)

\([[0]_3]\) has level \(\vartheta(\varepsilon_{\Omega+1})\)

\([[0]_4]\) has level \(\vartheta(\varepsilon_{\Omega_2+1})\)

\([[0]_{[0]}]\) has level \(\vartheta(\Omega_\omega)\)

\([[0]_{[0]_2}]\) has level \(\vartheta(\Omega_\Omega)\)

\([[0]_{[0]_{...}}]\) has level \(\psi(\psi_I(0))\)