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Extended definition:

1. If $$a$$ has label 0, we proceed as in Kirby-Paris' game. Call the node's parent $$b$$, and its grandparent $$c$$. First we delete $$a$$. If $$c$$ exists (i.e. $$b$$ is not the root), we make $$n$$ copies of $$b$$ and all its children and attach them to $$c$$.
2. If $$a$$ has label $$v + 1$$, (where v is the remainder of the label) we go down the tree looking for a node $$b$$ with label $$u \leq v$$ (which is guaranteed to exist, as nodes directly above root are all 0's). Consider $$b$$ and subtree with root at $$b$$, and call that subtree $$S$$. Create a copy of $$S$$, call it $$S_1$$. Within $$S_1$$, we relabel $$b$$ with $$u$$. Back in the original tree, replace $$a$$ with $$S_1$$. After that we replace the $$a$$ in $$S_1$$ with $$S_1$$ call it $$S_2$$. Within $$S_2$$, we relabel $$b$$ with $$u$$. After that we replace the $$a$$ in $$S_2$$ with $$S_2$$ call it $$S_3$$, etc. Go further until $$S_n$$. In $$S_n$$ we relabel a with 0.
3. If $$a$$ has label $$()$$, we simply relabel it with $$n + 1$$.
4. If $$a$$ has a hydra as label, we solve the hydra.

For clarity, we can write () for root and level 0 brackets and $$()_n$$ for level n brackets.

## Strength of Extended Buchholz Hydra

Hydra FGH
$$(()_n)$$ $$\vartheta(\Omega_{n})$$
$$(()_{()})$$ $$\vartheta(\varepsilon_{\Omega_{\omega}+1})$$
$$(()_{()1})$$ $$\vartheta(\Omega_{\omega+1})$$
$$(()_{()()})$$ $$\vartheta(\Omega_{\omega2})$$
$$(()_{(())})$$ $$\vartheta(\Omega_{\omega^2})$$
$$(()_{(()())})$$ $$\vartheta(\Omega_{\omega^3})$$
$$(()_{((()))})$$ $$\vartheta(\Omega_{\omega^\omega})$$
$$(()_{(((())))})$$ $$\vartheta(\Omega_{\omega^{\omega^\omega}})$$
$$(()_{(()_2)})$$ $$\vartheta(\Omega_{\varepsilon_0})$$
$$(()_{(()_3)})$$ $$\vartheta(\Omega_{\vartheta(\varepsilon_{\Omega+1})})$$
$$(()_{(()_{()})})$$ $$\vartheta(\Omega_{\vartheta(\varepsilon_{\Omega_{\omega}+1})})$$
$$(()_{()_2})$$ $$\vartheta(\Omega_{\Omega})$$
$$(()_{()_3})$$ $$\vartheta(\Omega_{\Omega_2})$$
$$(()_{()_{()_2}})$$ $$\vartheta(\Omega_{\Omega_{\Omega}})$$
$$(()_{()_{()_3}})$$ $$\vartheta(\Omega_{\Omega_{\Omega_2}})$$
$$(()_{()_{()_{()_2}}})$$ $$\vartheta(\Omega_{\Omega_{\Omega_{\Omega}}})$$
xBH(n) $$\psi_{I}(0)$$

## xBH(n)

$$R_1 = ()$$

$$R_x = ()_{R_{x-1}}$$

xBH(n) starts with the hydra $$((()_{R_x})))$$

## Array Hydra

Array Rules:

The active entry is the first non-zero entry.

1-4. Active entry is a hydra: solve like Extended Hydra to a row of ()'s

5. Zeroes at the end of an array must be removed

6. ◆,0,○+1,◆ = ◆,X,○,◆, where X is the smallest subtree with u<v where v is the array of the active bracket and u the array of another bracket, nested n times

### Array comparison

1. Is array a longer than array b? If yes: array a>array b

If no:

2. Is the last entry of array a bigger than array b? If yes: array a>array b, If no, look to the next entry

### Extended hydras

7. 0(a)b = n(a-1)n(a-1)n...n(a-1)n(a-1)n(a)b-1 with n n's

8. 0(())2 = 1(n)2 and other hydras work like they work in Extended Hydra exepted that the brackets that are nested 1 times solve to n.

## Strength of Array Hydra

I believe it is at least $$\psi_{\chi(M^\omega)}(0)$$ for linear arrays

For extended arrays, it goes beyond the limit of compact ordinals.

I should extend my notation, because this is even stronger than Dollar Function!

Hydra FGH
$$(()_{0,1})$$ $$\psi_{I}(0)$$
$$(()_{0,2})$$ $$\psi_{I}(1)$$
$$(()_{0,()})$$ $$\psi_{I}(I)$$
$$(()_{0,()()})$$ $$\psi_{I}(I2)$$
$$(()_{0,(())})$$ $$\psi_{I}(I^2)$$
$$(()_{0,((()))})$$

$$\psi_{I}(I^I)$$

$$(()_{0,(()_2)})$$ $$\psi_{I}(\varepsilon_{I+1})$$
$$(()_{0,(()_3)})$$ $$\psi_{I}(\varepsilon_{\Omega_{I+1}})$$
$$(()_{0,(()_{()})})$$

$$\psi_{I}(\varepsilon_{\Omega_{I2}})$$

$$(()_{0,()_2})$$ $$\psi_{I_2}(0)$$
$$(()_{0,()_3})$$ $$\psi_{I_3}(0)$$
$$(()_{0,0,1})$$ $$\psi_{I(1)}(0)$$
$$(()_{0,0,2})$$ $$\psi_{I(2)}(0)$$
$$(()_{0,0,0,1})$$ $$\psi_{I(1,0)}(0)$$
$$(()_{0,0,0,0,1})$$ $$\psi_{I(1,0,0)}(0)$$
limit of linear arrays $$\psi_{\chi(M^\omega)}(0)$$
$$(()_{0(1)1})$$ $$\psi_{\chi(M^M)}(0)$$
$$(()_{0(1)2})$$ $$\psi_{\chi(M^{M}2)}(0)$$
$$(()_{0(1)()})$$ $$\psi_{\chi(M^{M+1})}(0)$$
$$(()_{0(1)()()})$$ $$\psi_{\chi(M^{M+2})}(0)$$
$$(()_{0(1)(())})$$ $$\psi_{\chi(M^{M2})}(0)$$
$$(()_{0(1)(()_2)})$$ $$\psi_{\chi(\varepsilon_{M+1})}(0)$$
$$(()_{0(1)()_2})$$ $$\psi_{\chi(M_2)}(0)$$
$$(()_{0(1)0,1})$$ $$\psi_{\chi(M(1))}(0)$$
$$(()_{0(1)0,0,1})$$ $$\psi_{\chi(M(1,0))}(0)$$
$$(()_{0(1)0(1)1})$$ $$\psi_{\Xi(3,0)}(\Xi(3,0)^{\Xi(3,0)})$$
$$(()_{0(1)0(1)0(1)1})$$ $$\psi_{\Xi(4,0)}(\Xi(4,0)^{\Xi(4,0)})$$
$$(()_{0(2)1})$$ $$\psi_{\Xi(K)}(0)$$
$$(()_{0(3)1})$$ $$\psi_{\Xi(K^2)}(0)$$
$$(()_{0(())1})$$ $$\psi_{\Xi(K^K)}(0)$$
$$(()_{0((()_2))1})$$ $$\psi_{\Xi(\varepsilon_{K+1})}(0)$$
$$(()_{0(()_2)1})$$ $$\psi_{\Xi(K_2)}(0)$$
$$(()_{0(()_{0,1})1})$$ $$\psi_{\Xi(_20)}(0)$$
$$(()_{0(()_{0,2})1})$$ $$\psi_{\Xi(_30)}(0)$$
$$(()_{0(()_{0,()})1})$$ $$\psi_{\Xi(_\omega0)}(0)$$

$$(()_{0(()_{0,()_2})1})$$

$$\psi_{\Xi(_\Omega0)}(0)$$
$$(()_{0(()_{0,0,1})1})$$ $$\psi_{\Xi(_{1,0}0)}(0)$$
$$(()_{0(()_{0,0,0,1})1})$$ $$\psi_{\Xi(_{1,0,0}0)}(0)$$
$$(()_{0(()_{0(1)1})1})$$

$$\psi_{\Xi(_{\Xi(0)^\Xi(0)}0)}(0)$$

$$(()_{0(()_{0(1)()_2})1})$$ $$\psi_{\Xi(_{\Xi(0)_2}0)}(0)$$
$$(()_{0(()_{0(1)0,1})1})$$ $$\psi_{\Xi(_{\Xi(1,0)}0)}(0)$$
$$(()_{0(()_{0(1)0(1)1})1})$$ $$\psi_{\Xi(_{\Xi(1,0)^\Xi(1,0)}0)}(0)$$
$$(()_{0(()_{0(2)1})1})$$ $$\psi_{\Xi(_{\Xi(K)}0)}(0)$$
$$(()_{0(()_{0(())1})1})$$ $$\psi_{\Xi(_{\Xi(K^K)}0)}(0)$$
$$(()_{0(()_{0(()_{0,1})1})1})$$ $$\psi_{\Xi(_{\Xi(_20)}0)}(0)$$
$$(()_{0(()_{0(()_{0(()_{0,1})1})1})1})$$ $$\psi_{\Xi(_{\Xi(_{\Xi(_20)}0)}0)}(0)$$
limit of extended Array Hydra. $$\psi_{U}(0)$$, the limit of extended compact ordinal notation.