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blashimov comments on Standard and Nonstandard Numbers - Less Wrong
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I'm going to have to read the proof of the hydra game, because I pretty quickly got over 2.8k nodes and still in increasing...
It's even worse than that, depending on how you start, you can easily get 100s of thousands of nodes...
It's even worse than that, the function for the maximum number of nodes you end up before they start going down, if you play using the worst possible strategy, increases faster than any function which Peano arithmetic can prove to be total (i.e., it grows faster than any Turing machine run on various inputs, which Peano arithmetic can prove to halt for any input). To say that this grows faster than the Ackermann function is putting it very mildly.