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) \(\psi_{I}(0)\)


\(R_1 = ()\)

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

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


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


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


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


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

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