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That's why this is about the https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice
by belovedeagle 12y ago
That's why this is about the https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice
- couchand 12y agoI'm not clear on how the Axiom of Choice helps here. So you want a choice function, and you want it without enumerating every set. Great, the AoC gives you that. Now, you want to execute that choice function on every equivalence class. The AoC says nothing about that - you still need to be able to enumerate every set you would like to choose from. Giving all the prisoners the choice function is by itself useless: once they observe the sequence, they can tell which equivalence class they're in. That's a tautology. For the suggested strategy to work, the prisoners need not only the choice function (which they get by the AoC), but an actual element from every set. It doesn't seem like the Axiom of Choice is relevant for that part.
- belovedeagle 12y agoI'm still not clear on what the issue is. Like you said, the prisoners observe the sequence and thus know which equivalence class they're in. The choice function (thanks to AC) gives them an "actual element" from that set.
- couchand 12y agoI'm saying it's incorrect to suggest they can enumerate a representative example of every equivalence class simply because they have a choice function. Of course they can pick an element from any of the equivalence classes in bounded time (by the Axiom of Choice) but that says nothing about picking an element from every one of those sets in bounded time.