What I don’t get is that for each and every sequence, there exists an identical sequence with a different starting hat. So you can never know which EC you’re in, only which EC everyone in front of you is in. So you can never know what to say.

What am I missing? Please help…

(Btw Stan I love the link you posted on May 24!)

]]>For instance

101001000… may appear to be equivalent to 010010001…

for the first person but not to the second person

I am not sure if this changes anything but it is another weird aspect to this.

Also, one thing that isn’t clear is if the prisoner’s no their position in line. When guessing are they do as if they are in the first position or in the same corresponding position that they actually are in?

Lastly, isn’t each person’s guess irrelevant? Since any person who can see an infinite part of the sequence in front of them must be in the finite part? Or do these finite difference not necessarily occur at the beginning?

]]>The theorems are all of the form “a strategy exists”, as opposed to: “the prisoners can formulate some sort of strategy telling them how to behave depending on the hats that they see”. A strategy is a certain set-theoretic object. ZFC asserts ONLY that this object exists as a set (not that it can be “found” or used as an actual algorithm procedure).

]]>And of course these are not real people but abstract math “people” but it still remains that the point of the problem is to find a precedure.

]]>Stan Wagon

]]>The thing is, is doesn’t matter which one you take. Your algorithm could be “Pick a random one, move on to the next set”. Of course one might ask “How does one pick a random element?” I would argue that while I can not explicitly say how, it is clear that that is possible in the same way that I can pick a random pebble out of the ocean.

]]>lots of mathematicians do, actually. And this is not just a “bunch” of sets, it’s uncountably many of them, and there is no algorithm at all that can tell you which representative you should take (and by that i mean that it’s impossible to create an algorithme to do that, it can’t exist one) and yet, by the axiom of choice, you get to choose a representative from every set.

]]>The part that is so hard to wrap your head around is that the prisoner can look out upon an infinite sequence and tell which equivalence class it comes from. This would obviously be impossible in the real world but in the world of perfectly intelligent prisoners, it is fine. Not only is it fine but it does not use the axiom of choice at all.

The axiom of choice comes in just to make it possible that there is a representative sequence for the equivalence class. I don’t think anyone would find it far fetched that you can choose a representative from a bunch of sets.

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