Buchholz hydra

Buchholz hydra game is a one-player game similar to Kirby-Paris hydra game. As opposed to latter game, Buchholz's game uses trees labelled with any finite number or \(\omega\).

Rules
Game starts with any labelled tree T such that root is marked with special + label and every child of root has label 0. This tree is called hydra. At each step we choose one head of hydra (i.e. node without any children) together with a number n and we change hydra using one of following rules: Let's denote hydra after applying this rule to rightmost head T(n)
 * 1) If head we chose has label 0, we proceed as in Kirby-Paris' game, that is we delete that head and we go one level down into the hydra. If node we are actually at is root, we do nothing. Otherwise, we make n copies of subtree starting at that node and we connect them to even lower node.
 * 2) If head we chose has label u+1, we go down the tree looking for node \(\alpha\) with label v≤u (which is guaranteed to exist, as nodes above root are all 0's). Let's call subtree with \(\alpha\) as root T'. Now we replace head we chose with T' modified as follows: node \(\alpha\) now has label u and chosen head now has label 0.
 * 3) If head we chose has label \(\omega\), we simply replace that label with n+1.

We call a sequence of heads we choose and n's we choose a strategy. Strategy is called winning, if by following this strategy after finitely many steps we are eventually left with root node only, and strategy is losing otherwise.

Buchholz's results
Buchholz gave three theorems concerning his hydra game: Specifically, given path tree \(T^n\) with labels \(+,0,\omega,\omega,...,\omega\) it can't be proven that for all n there exists k such that \(T^n(1)(2)(3)...(k)\) is root tree.
 * 1) There are no losing strategies for any hydra
 * 2) For any fixed hydra above statement is provable in \(\Pi_1^1-CA_0+BI\)
 * 3) First theorem is unprovable in \(\Pi_1^1-CA_0+BI\)

Termination function
Define function BH(n) as smallest k such that \(T^n(1)(2)(3)...(k)\) as defined above is root tree. By theorem 1 this function is well-defined, although, by theorem 3, its totality can't be proven within  \(\Pi_1^1-CA_0+BI\). It's rate of growth is comparable to \(f_{\psi_0(\varepsilon_{\Omega_\omega + 1})}(n)\)