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View Full Version : Question for Math People: Axiom of Choice Good or Bad



VTBoy
2004-Dec-02, 01:19 AM
What is your opinion of the Axiom of Choice. I find the Banach-Tarski paradox and WOP in the Reals to be very disturbing property of the Axiom of Choice. So whats your opinion of it.

Severian
2004-Dec-02, 03:43 PM
It does allow some strange results, but I'm still in favor of it. The statement of the axiom that I like the best is "Any arbitrary product of nonempty sets is also nonempty" (even infinite). But I know people that don't like it, and I respect that, and I think most people do try to note if they are using it. There are some pretty slick results that use it in an essential way; I like the results, so I like the axiom ;)

Gerrsun
2004-Dec-02, 04:00 PM
It's got a nice beat but you can't dance to it, so I give it a 7. :P

martin
2004-Dec-02, 07:39 PM
What is your opinion of the Axiom of Choice. I find the Banach-Tarski paradox and WOP in the Reals to be very disturbing property of the Axiom of Choice. So whats your opinion of it.

So you are not learning with all the trouble you are starting with 0.999..., so we must defend real number system from attack :D

It is a long time since I am thinking on these things. It can be used to show existence of subsets of real line that are not measurable under system of Lebesgue measure. In more general, there are impossibility theorems about assignment of non-trivial measure to sigma-algebra on uncountable sets using axiom of choice. (I cannot remember what it is called, when every subset is measurable.) But maybe there are other ways to prove these...

Martin

Disinfo Agent
2004-Dec-02, 08:20 PM
Well, rejecting the AC makes life considerably more difficult for mathematicians:


The full strength of the Axiom of Choice does not seem to be needed for applied mathematics. Some weaker principle such as CC or DC generally would suffice. To see this, consider that any application is based on measurements, but humans can only make finitely many measurements. We can extrapolate and take limits, but usually those limits are sequential, so even in theory we cannot make use of more than countably many measurements. The resulting spaces are separable. Even if we use a nonseparable space such as L^(infinity), this may be merely to simplify our notation; the relevant action may all be happening in some separable subspace, which we could identify with just a bit more effort. (Thus, in some sense, nonseparable spaces exist only in the imagination of mathematicians.) If we restrict our attention to separable spaces, then much of conventional analysis still works with AC replaced by CC or DC. However, the resulting exposition is then more complicated, and so this route is only followed by a few mathematicians who have strong philosophical leanings against AC.

Axiom of Choice (http://www.math.vanderbilt.edu/~schectex/ccc/choice.html)