Jarhyn
Wizard
- Joined
- Mar 29, 2010
- Messages
- 14,506
- Gender
- Androgyne; they/them
- Basic Beliefs
- Natural Philosophy, Game Theoretic Ethicist
Well that's the thing, Swammerdami, I don't think randomness is unrelated to the problem at all, for the reasons I articulate. I think that the AC has a fundamental assumption built into it about the nature of solvability and randomness and information derivable from infinites implied by this equivalence class that you describe.The other botch I made was presenting the setup as though each prisoner sees a finite number of hats instead of an infinite number.
~ ~ ~ ~ ~ ~ ~
@ Jarhyn — I did notice that you were pointed at the solution, but you didn't articulate it completely. And you insisted on randomness, which is irrelevant to the puzzle.
I think you were arguing that solution is impossible, which it is (without the Axiom of Choice). But David Hilbert said "Mathematics is a game played according to certain simple rules with meaningless marks on paper." Play the game!
The AC essentially posits that randomness is contradictory and thus all sequences expressed have a finite process description, even if the sequence itself is infinite, and in fact it can only be resolved if the sequence is infinite!
Either the prisoners or the wardens have an impossible task, and it all hinges on whether you can express infinite randomness.
Edit: and as I think about it, this would equate to whether the set theory allows expression of U as a concept: a description of all descriptions would need to describe itself...
Largely, this is seen as a contradiction in rigorous constructions as I understand it, that U is a made-up nonsense idea that if it can be expressed means a trivialization of axioms?
So I would reject that randomness can possibly be expressed, and the prisoners actually do win, and the AC holds because an assumption of randomness means the experiment is nonsense.