Is it really impossible to lose the Hydra game?

ordinals graph-theory provability

Solution 1:

Your description of how the hydra "regenerates" is incorrect. You don't add copies of the entire tree above the grandparent node, only part of the tree above the grandparent node that starts with going to the parent node. This means that after cutting off a head from $((xx)(xx))$, you get $((xx)(x)(x))$, because you only duplicate the $(x)$ part you had left after chopping off the head, not the other $(xx)$ from which you didn't chop off a head.

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