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A) Very interesting, informative, I feel a tad smarter. :P

B) The people talking about time are confusing math and reality. Math at this level exists outside of the known physical constructs we encounter – this is where the ‘1 true topos’ argument comes from, which is interesting in that if there is no ‘true’ topos, what, exactly, is physicality? At any rate it should be fairly obvious that this whole question of time and memory is bogus and a limit of your imagination. We think time at any rate is a by-product of physical space, memory or information is probably also (we think) a by-product of space, not something inherent to math.

C) Viewed from the concept of an information problem, I think this is also very interesting. I don’t fully understand the Axiom of Choice, but at least from what some people were saying about probabilty, I think I may have spotted a fallacy. They are presuming each guess is independent – they are dead wrong.

It is true that each prisoner’s guess isn’t dependent in reverse, on the previous choice, but they have visibility of ‘all but a finite set’. They have infinite dependency upon future elements.

So, basically, we have an infinite track tape and the warden is choosing some finite part of that track tape to screw up, and afterwards at some point in the future the assertion is that seeing infinity allows infinite numbers of correct choices on that track tape.

I’ll admit, I don’t understand how that works, and that may be that I don’t understand the equivalence classes very well but:

Are we asserting that all infinite equivalence classes are the same?

Why can’t the warden make infinitely many choices? The prisoners are choosing infinitely many times, I see no reason presented the warden cannot.

So, I’m highly tempted to say the problem presented is bogus, not the Axiom of Choice.

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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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