The Axiom of Choice is Wrong (2007)

So this guy is against using the axiom of choice because a countably infinite number of prisoners in a row who have agreed on an infinite set of equivalence classes just between breakfast and hat game time and remember them and each of them can recognize infinitely many hat colors in finite time (takes deep breath) suddenly becomes unintuitive using the axiom of choice?<p>I can't help him there.

Be sure to check out the comments. Lots of good discussion in there, on both the paradox at hand and mathematical Platonism in general, by people who know what they're talking about.

More troubling to me has been the continuum hypothesis. The notion that you can toss a dart (whose tip is the width of a point) at the real number line and hit an integer or rational number with probability zero is very unintuitive.<p>I can kind of grasp the continuum hypothesis, there does seem to be a distinction between countable and uncountable infinity. Continuing the game beyond that to this area of large cardinals strikes me as just a language game.<p>Most mind blowing, IMHO, is Cantor's middle third's set[1]. Uncountable nowhere dense totally disconnected, ... it's an amazing set and so easy to construct.<p>[1] <a href="http://en.wikipedia.org/wiki/Cantor_set" rel="nofollow">http://en.wikipedia.org/wiki/Cantor_set</a>

&#62; Its interesting to notice that a larger number of hat colors poses no problem here. For any set of hat colors , the prisoners can pick an abelian group structure on . Then, the first prisoner guesses the ’sum’ of all the hat colors he can see. The next guy can then subtract the sum of the hat colors he sees from the hat color the first guy said to find his own hat color. Again, this argument repeats, and so everyone except the first guy gets out. For the case of black and white, the previous argument used black = 0 (mod 2) and white = 1 (mod 2).<p>So the first guy says the sum, not a color? If he's allowed to say arbitrary things he can as well tell everyone their color "next guy is white, next green, etc.".

This is still only an argument against the uncountable axiom of choice.<p>The set of equivalence classes of infinite strings is uncountable, so using the axiom of choice here is invalid.<p>As far as I know, the countable one is still reasonable.

I feel like the axiom of choice can't be used here.<p>Let's say we have a infinitely long bit string, b. This equivalence class has an infinite number of elements in it! That is, there is an infinite number of strings with suffix b.<p>b, 0b, 1b, 00b, 01b, 10b, 11b, etc.<p>Can you use the axiom of choice in this case? I think it is required that each bin has a finite number of objects, even though there are infinite bins.

Looks like another example of infinity leading to nonsensical (or at least unintuitive) results.

If you choose not to decide, you still have made a choice. -- unsourced René Descartes

Sounds like Bridge. I hate Bridge!