I have been researching regarding the famous hat puzzle, and all the solutions I have found seem incorrect to me.

They all talk about counting hats of a certain color and responding depending on if that number is odd or even.

But NOWHERE in the puzzle it is written that the number of white and black hats is THE SAME! They may have 81 black and 19 white hats, for example. How do you solve that?

Cheers!

]]>Thank you very much for the link. My concerns with the infinite hats problem are very similar to the ones expressed by David Wilson’s comment there.

I also liked the blog itself (The Everything Seminar) and added it to my blogroll.

]]>I like though the solution with the axiom of choice better: If you are dealing with infinity you have already left the mundane world and work with the abstraction. It is an old controversary in Mathematics if one should use the axiom of choise or not. As far to my knowledge there exists some mathematician who restrict themselves only to what can be proved without it. If one take on the Bourbaki’s position – the theory based on a subset of axioms have to be a subset of the theory based on the whole set. So, by excluding axiom of choice you get the theorems which are valid with axiom of choice (as far to my knowledge there is even some activity in keeping track which theorems require axiom of choice, and which are independent of it).

I, personally, find the axiom of choice very practical, as I consider my speciallity to be construction of counterexamples, and in that light the only impotant fact about the object constructed is that it exists.

What for the development of Mathematics – it is a matter of definitions: Where does it ends and Computer Sciences or Physics starts? There always be people who derive pleasure of the “mental acrobatics”, so I don’t think this part of Mathematics will die away (and I think it is a good thing).

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