# Phase Transition Thresholds for Generalized Goodstein Sequences, Hydra Games and Ackermannian Functions.

## Gabriele Buriola

**Authors:** Gabriele Buriola and Andreas Weiermann

We extend previous results [2] regarding the phase transition thresholds for Goodstein Sequences and Hydra Games to a more general setting. In both cases, we substitute the original successor function with the iterations of a strictly increasing primitive recursive function \(g\) satisfying the condition \(g(x) \geq x+1\); more precisely, the steps of the Hydra Game, originally of type \(\alpha_{f\!,i+1}= \alpha_{f\!,i}[1+f(i)]\), are now of the form \(\alpha^{f\!,g}_{i+1}=\alpha^{f\!,g}_{i}[1+f(g^{i-1}(1))]\), while the steps of Goodstein sequences are changed from \(m_{f\!,i+1}=m_{f\!,i}\left(1+f(i) \mapsto 1+f(i+1)\right) -1\) to \(m^{f\!,g}_{i+1}=m^{f\!,g}_i\left(1+f(g^{i-1}(1)) \mapsto 1+f(g^{i}(1))\right) -1\). The new phase transition thresholds incorporate the starting function \(g\). These findings also allow a generalization of the phase transition threshold for Ackermannian functions [1] and fit within a rich literature regarding phase transitions in provability [3,4,5].

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