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If you were to actually operationalize this in the context of the problem, then what you would find is that for all of the halting Turing machines, prisoner “n” (starting from the first one who must call out a color) would have to say, “my Turing machine halts”. The first prisoner who corresponds to a non-halting Turing machine will attempt to simulate, in his head, a Turing machine that runs forever, and he won’t be able to call out a color until the simulation in his head finishes. It won’t finish, and the warden and all prisoners will die before he calls out a color.

[Nonsense warning] If “w” = Omega, then worst case we have to wait “w*w” time for the prisoners to give answers, assuming we go in sequence and (patiently) wait possibly an infinite amount of time for each one to answer. I’m not an expert on hyper-computation, but my understanding is that computable power can be indexed by ordinals.

I’m assuming here for simplicity that (1) a particular Chaitin constant has been selected and (2) by chance, the hats of ALL of the prisoners (not just all but finitely many) are identical with the true binary representation of this number. So if only they had the time, they could ALL go free.

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