User blog:Wythagoras/Upper bound on the Buchholz hydra

I have to change the hydra rules a bit for a easier analysis for me. It spawns n nests instead of one, so it only makes the hydra stronger. (n is the number of steps that has been done, including the step itself to make sure it's at least one)

First, we look to the hydra (+(0(1))) it spawns (+(0(0(0)))) in the normal version, so it'll reduce in 38 steps. In the other version, (0(1)) can be upper bounded by f(ε0), as it spawns n nests.

(0(1)(0)) expands to (0(1))(0(1))...(0(1))(0(1)), so it can be upper bounded by f(ε0+1).

Again, we get a table: Ι believe it's enough, we already reached (0(1)(1)(1)(1)), the hydra for BH(3), at ε3.

We can say \(\text{BH}(3) < f_{\varepsilon_3}(6)\).