A function f exists that maps from other people’s hat colors and the prisoner’s own ID to a color guess, such that the application of this function will only result in finite failures, independent of the particular hat coloring.

But just because the strategy “is out there”, it doesn’t mean that the prisoners can utilize it or even know whether they are utilizing it. The book on the cure of cancer is out there in the Library of Babel (all possible 1000 page books), the actual hard part is to tell it apart from all other similar books.

It boils down to the ‘invent vs discover’ problem or the ‘create vs choose’ problem. Every creation is basically choice out of a sufficiently large set of possibilities. When you invent something you discover that particular construction from the set of all possible constructions. Whenever you write a poem or novel you basically choose one from Library of Babel. I wouldn’t say that the Mona Lisa existed before Da Vinci painted it, even though that configuration of pixels already ‘existed’ before (or configuration of the quantum states of elementary particles).

The platonic existence concept only describes potentials, things that might as well happen that way but nobody can make them actually happen. It’s like hitting Chaitin’s omega number over the number line by throwing a dart. It’s not impossible to hit it, since “the number exists”, but you cannot aim for it because you don’t even know it precisely in the real world. If you could invoke hypercomputation then you could hit it.

For me any existence claim is inherently coupled with a (potentially infinite or otherwise weird) construction process, and the existence is only valid to the same extent that I accept the construction ‘story’ as valid.

The only (but huge) difference is that the prisoners don’t need to be intelligent or creative. They just need to be able to make an infinite amount of agreements and then remember the agreements. They may do it in some way where they spend 1 second on it in total and they make a decision at each real number seconds or something along these lines. But the have to remember an infinitely complex agreement (with such memory they could even hold the ‘halting function’ in their head).

I have no problem to accept the solution. If they have the means to go through an infinity of arbitrary decisions and have infinite memories, then sure they can win at this game.

In practice however, the guards wouldn’t notice a thing. The prisoners would look like random players and neither the prisoners nor the guards could know an upper bound on when the block of infinite correct guesses will begin. It will “at some point”, but it’s an unfalsifiable claim. Any finite sequence of guesses that looks random is totally compatible with the prisoner’s claim that they have this great strategy.

]]>Why Stuff is Hard | The Everything Seminar

]]>First of all sorry for my bad English. Your three posts about divergent series is excellent! I found a way to sum divergent series and determine the limits of function in their singular points. I discovered that the method can be applied to compute divergent integrals.

Sincerely, Sinisa

]]>The 8th comment down is a great refutation of the situation proposed in the post. ]]>

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