User blog comment:Vel!/Nested Buchholz hydra/@comment-5529393-20131224220731

Actually, the strength of the notation is $$ \theta (\Omega_\Omega, 0) $$. A notation with hydras labelled up to limit ordinal $$\alpha$$ has strength $$\theta (\Omega_\alpha, 0) $$. So if we let H_0 be hydras labelled by natural numbers, then it has ordinal $$S_0 = \theta (\Omega_omega, 0)$$. If we let H_1 be hydras labelled with hydras from H_0, it has ordinal $$ S_1 = \theta(\Omega_{\theta (\Omega_omega, 0)}, 0)$$, and so on. The limit of this procedure will be the first fixed point of $$f(\alpha) = \theta (\Omega_\alpha, 0)$$, which is $$\theta (\Omega_\Omega, 0)$$, not $$\psi (\psi_I (0))$$.