## FANDOM

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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 $$S(T^\omega)$$, $$S(T^T)$$, $$S(\varepsilon_{T+1})$$