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

Which rule should you use?

  1. If there is nothing after the $, use rule 1
  2. If there are any non-nested non-subscript numbers, use rule 2
  3. If there are any non-nested non-subscript [0]'s, use rule 3
  4. If there are any non-nested non-subscript [b]'s, use rule 4
  5. 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
  6. 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 \(S(T^\omega)\), \(S(T^T)\), \(S(\varepsilon_{T+1})\)

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