Dollar function: final version

Inspired by Hyp cos I decided to remake everything exepted for Bracket Notation.

Which rule should you use?

If there is nothing after the $, use rule 1
If there are any non-nested non-subscript numbers, use rule 2
If there are any non-nested non-subscript [0]'s, use rule 3
If there are any non-nested non-subscript [b]'s, use rule 4
If the previous things doesn't apply but the lowest level bracket can be solved with normal bracket notation:
1. Search in the bracket for the least nested lowest level bracket or number
  1. If it is a 0:
    1. If the zero is the only content, use rule 3
    2. Otherwise, use rule 5
  2. If it is another number, use rule 4
  3. If it is a bracket: Return to step 5
If the lowest level bracket can be solved with extended bracket notation:
1. Is the number in the typed bracket a 0, use rule 6 and the subrule if needed
2. Otherwise, use rule 4

$\bullet$ can be anything

$\circ$ is a group of brackets

$\diamond$ is a group of zeroes

Extended Bracket Notation

This works now a bit like the Buchholz hydra, and the limit is $\psi(\psi_I(0))$

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.

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.

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 $\vartheta(\varepsilon_{\Omega+1})$

Linear Array Notation

Here are no $, but this are just rules what you should do if that kind of array is the lowest level array.

7. $[[b\bullet,c]] = [[0,c-1]_{[b-1\bullet,c]1}]$

8. To diagonalize in the nth position with bracket types, you must use $[\underbrace{0,0...0,1}_n]_k$ They diagonalize in the last entry, for any type of bracket.

9. $[[\diamond,b\bullet,c,\bullet]] = [[\diamond,[\diamond,b\bullet,c-1,\bullet]_{[\diamond,b-1\bullet,c,\bullet]},c-1,\bullet]$

10. $[0,c,\bullet] = [0]$

S3. Zeroes at the and of the array must be removed

Analysis

$[[[0],1]]$ has level $\psi(\psi_I(0))$

$[[[0][0],1]]$ has level $\psi(\psi_I(1))$

$[[[1],1]]$ has level $\psi(\psi_I(\omega))$

$[[[[0]_2],1]]$ has level $\psi(\psi_I(\varepsilon_0))$

$[[[[[0]_2]_2],1]]$ has level $\psi(\psi_I(\zeta_0))$

$[[[[[0]_3]_2],1]]$ has level $\psi(\psi_I(\varphi(\omega,0)))$

$[[[[[0],1]],1]]$ has level $\psi(\psi_I(\psi(\psi_I(0))))$

$[[[[[[[0],1]],1]],1]]$ has level $\psi(\psi_I(\psi(\psi_I(\psi(\psi_I(0))))))$

$[[[0]_2,1]]$ has level $\psi(\psi_I(\Omega))$

$[[[0]_{[0]},1]]$ has level $\psi(\psi_I(\Omega_\omega))$

$[[[[0],1],1]]$ has level $\psi(\psi_I(\psi_I(0)))$

$[[[0,1]_2,1]]$ has level $\psi(\psi_I(I))$

$[[[1,1]_2,1]]$ has level $\psi(\psi_I(I\omega))$

$[[[[0]_2,1]_2,1]]$ has level $\psi(\psi_I(I\Omega))$

$[[[[0,1],1]_2,1]]$ has level $\psi(\psi_I(I\psi_I(0)))$

$[[[[0,1]_2,1]_2,1]]$ has level $\psi(\psi_I(I^2))$

$[[[[1,1]_2,1]_2,1]]$ has level $\psi(\psi_I(I^\omega))$

$[[[[[0,1]_2,1]_2,1]_2,1]]$ has level $\psi(\psi_I(I^I))$

$[[[0,1]_3,1]]$ has level $\psi(\psi_I(\varepsilon_{I+1}))$

$[[[[[0],1]_3,1]_3,1]]$ has level $\psi(\psi_I(\varphi(\omega,I+1)))$

$[[[[[0,1]_3,1]_3,1]_3,1]]$ has level $\psi(\psi_I(\Omega_{I+1}))$

$[[[0],2]]$ has level $\psi(\psi_{I_2}(0))$

$[[[0],[0]]]$ has level $\psi(\psi_{I_\omega}(0))$

$[[[0],[0]_2]]$ has level $\psi(\psi_{I_\Omega}(0))$

$[[[0],[0,1]]]$ has level $\psi(\psi_{I_{\psi_I(0)}}(0))$

$[[[0],[0,2]]]$ has level $\psi(\psi_{I_{\psi_{I_2}(0)}}(0))$

$[[[0],[0,1]_2]]$ has level $\psi(\psi_{I_{I}}(0))$

$[[0,[0],1]]$ has level $\psi(\psi_{\chi(1)}(0))$

$[[0,[0][0],1]]$ has level $\psi(\psi_{\chi(1)}(1))$

$[[0,[0]_2,1]]$ has level $\psi(\psi_{\chi(1)}(\Omega))$

$[[0,[0,0,1]_2,1]]$ has level $\psi(\psi_{\chi(1)}(\chi(1)))$

$[[0,0,2]]$ has level $\psi(\psi_{\chi(2)}(0))$

$[[0,0,3]]$ has level $\psi(\psi_{\chi(3)}(0))$

$[[0,0,[0]]]$ has level $\psi(\psi_{\chi(\omega)}(0))$

$[[0,0,[0]_2]]$ has level $\psi(\psi_{\chi(\Omega)}(0))$

$[[0,0,[0,1]]]$ has level $\psi(\psi_{\chi(\psi_I(0))}(0))$

$[[0,0,[0,0,1]_2]]$ has level $\psi(\psi_{\chi(M)}(0))$

$[[0,0,[0],1]]$ has level $\psi(\Psi_{\Xi(3,0)}(0))$

$[[0,0,0,[0],1]]$ has level $\psi(\Psi_{\Xi(4,0)}(0))$

limit of linear arrays is $\psi(\Psi_{\Xi(\omega,0)}(0))$

Extended arrays

Entry counter

11. $[0]e^c0 = [0]$

12. $[0]e(b\bullet) = [0,0...0,1]e(b-1\bullet)$ a entries

limit of the entry counter is $\psi(\Psi_{\Xi(K)}(0))$

Dimensional arrays&Extended entry counter

11. $[0]e^c0 = [0]$

12. $[0]e^c(b\bullet) = [0(c)0...0(c)1]e^c(b-1\bullet)$ a entries

13. $[\diamond(c)b] = [\diamond(c)b-1]e^c([\diamond(c)b-1]e^c([\diamond(c)b-1]e^c(...)))$ where the $e^c$ operator works on the first dimension before (c) and there are no lower dimensions in $\diamond$

An comma is can be written shorthand for (0).

limit of dimensional arrays is $S(T^\omega)$.

Nested Array Notation

14. $[b\bullet(\diamond,0,c)1] = [0(\diamond,[b-1\bullet(\diamond,0,c)1][b-1\bullet(\diamond,0,c)1],c-1)1]$

limit of nested arrays is $\varepsilon_0$ ordinal structure.

The S function

In the first row, we have $\Omega$, $I$, $M$,$\Xi(3,0)$, ...

In the second row, we have $K$, $\Xi_2(1,0)$, $\Xi_2(2,0)$, ...

It is inaccessible cardinal diagonalizing over $K$ instead of $\Omega$

After that, we can also have a third row, a forth row, a plane, up to n dimensions.

Then we can have a diagonalizer over the dimensions, which can be a new dimension. Of course be can extend it to rows of dimensions, n-dimensions of dimensions, n-dimensions of dimensions of dimensions, ...

A function from Hyp cos: $S(\alpha,\beta,\gamma,\delta,...)$ = The $\alpha$th ordinal in the $\beta$th row in the $\gamma$th plane in the $\delta$th 3-dimension...

This is like the Velben hierarchy, so you can extend it with $S(T^\omega)$, $S(T^T)$, $S(\varepsilon_{T+1})$