godlessmath

I know a handful of people here are mathematically inclined, so let me tell you guys a story. My atheism came out today to some of the other grad students here, in an amusing way. I was talking to them about the axiom of choice. I usually like to tell people that I don't believe it is true, that I can buy the countable version but not the uncountable one.

So we were contemplating the reasonableness of the axiom of choice, when one of them said "let me put it this way, do you believe in God?" I said, actually, no I don't. He was surprised, and then said "well, I was going to say that if you do, then it's the same thing."

So what do you fellow logicians-mathematicians say? Should we have achoosism, the lack of belief in the axiom of choice?

Silent Acorns

I'm just an engineer, so I'm not sure if I qualify "mathematically inclined" by your standards <img src="graemlins/notworthy.gif" border="0" alt="[Not Worthy]" />

But I'll give it a shot ...

I had to look up what the "axiom of choice" is ... sounds so simple and obvious at first doesn't it? ... just like god did when I was a little kid ...

It makes me ask this question: if god was countable, would he still be worthy of worship?

I'd have to say no.

[ October 25, 2002: Message edited by: Silent Acorns ]</p>

godlessmath

Ha! Good point, probably not. Of course, if he were a finite set, would he even be a god?

In any case, the way I see it, if you have a list of sets, then yes, you can pick a guy from each set. You just go one by one, and if you were Zeus, you could make all your choices in under a minute. You pick the first element from the first set in 1/2 a minute, pick the second guy from the second set in 1/4 of a minute, pick a guy from the third set in 1/8 of a minute, and so on.

But any arbitrary collection of sets? Well...

Friar Bellows

Could someone please explain to me what the hell the axiom of choice is in words that wouldn't scare an ex-physical scientist? Especially why it's so important or such an object of debate. I went to MathWorld and read this:

"[The axiom of choice] states that, given any set of mutually exclusive nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets."

The above seems bloody obvious, but probably I don't really understand it.

wade-w

The Axiom of Choice doesn't include a "mutually exclusive" condition. But remove that clause, and that's the way its usually thought of.

More formally, it states that for any given collection of non-empty sets C, there exists a function f such that for each set S in C, f(S) is a member of S. This f is called a choice function. What this means is there is some method for making the choices.

And you're right. It does seem very obvious. But problems begin to develop when we start considering collections of sets like all non-empty subsets of the real numbers. It is not at all clear how to choose f is this case.

To make matters worse, it turns out that the Axiom of Choice is logically equivalent to the Well Ordering Principle, which is far from obvious.

The well ordering principle states that every non-empty set can be well ordered. How do you go about choosing a well order on the real numbers?

Also, there are some very strange results that can be proven using the Axiom of Choice. For example, consider the <a href="http://www.wikipedia.org/wiki/Banach-Tarski_Paradox" target="_blank">Banach-Tarski Paradox</a>. This basically says that you can split up the solid 3 dimensional unit ball into a finite number of slices and using only translation and rotation, reassemble the pieces into two balls, each having the same volume as the original. Note that the use of the word paradox in the name of this theorem is used not in the logical sense, but is an indication of how incredibly counter intuitive it is.

As for its importance, it (or the well ordering principle) is required for the proofs of several key theorems from many branches of mathematics, ranging from linear algebra to functional analysis to topology.

[ Edited because I was half asleep when I wrote this ]

[ October 26, 2002: Message edited by: wade-w ]

[ October 26, 2002: Message edited by: wade-w ]</p>

wade-w

I think most mathematicians who accept the Axiom of Choice do so mostly because it makes things easier. There are so many results in so many fields that you'd have to throw away, that they just grin and bear it. I guess it really boils down to: Are you a constructivist?

[ October 26, 2002: Message edited by: wade-w ]</p>

godlessmath

Constructivist? No not really. That can get annoying after a while. However, there are some results out there that I just can't see being proved without using the axiom of choice. Thus, I am afraid that sometimes the axiom isn't being used out of convenience, but out of necessity.

Which I suppose, since it IS an axiom. However, does the axiom of choice reflect reality? And if not, is our use of it reducing mathematics to manipulating symbols which may say false things about the universe?

wade-w

I agree that it is sometimes used out necessity. Thats what I meant when I said that there are many key results that would have to be thrown out without it. Imagine not being able to say that every vector space has a basis!

As far as "reflecting reality", not all mathematicians think that's a requirement. Consider the parallel postulates. Which of the three choices reflects reality?

Someone once said:
He was joking, of course. But my intuition follows this closely. Of course, that tells us what intuition is good for!

From all of this, I guess you could say I have mixed feelings on the subject. I think that it is prudent to use the weaker versions whenever possible, and resort to the full blown axiom of choice only when there is no other alternative. In other words, I'd have to say I'm an agnostic with respect to the axiom of choice.

godlessmath

Fair enough. I have a hard time buying that every proper ideal is contained in some proper maximal ideal. This is proved using Zorn's Lemma. It's too good to be true.

I remarked to someone that it is not an easy task to squeeze meaningful results from a handful of true statements. However, if you make a mistake and acccidently "prove" something that is false, then it seems that you can prove a plethora of statements after this.

Ernest Sparks

I am of another mind on this. I badly want to have useful macho numbers (transfinites), so I am willing to accept a full blown AC axiom. It is important to remember that it is an axiom and therefore freely chosen. Once accepted, then one must accept all the consequences. I don't blame anyone that is skeptical of AC and its equivalents, and it is a good idea to keep an eye on it.

I must add that I was a math graduate, but not a math professor. The only real "mathematician" I ever knew was in my class and took a civilian job with the US navy, with a job title of "Mathematician". He programmed computers for them.

[If you really want it, you can get it. That's math! But watch out what you wish for. That's life!]