Do you want the Axiom of Choice?

[december]

This topic seems like a natural extention of the thread on whether mathmatics is made up.

Axiom of Choice: Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.

Bertrand Russell…once said, “To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.”

The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm – e.g., “always choose the left shoe.” Why does Russell’s statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed – we can choose one member of each pair using the definition of “nonempty,” and we can repeat an operation finitely many times using the rules of formal logic.

http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

Paul Cohen used technique called “forcing” to prove the independence in set theory of the axiom of choice and of the generalised continuum hypothesis.

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Cohen.html

Cohen’s result tells us that we can choose to develop mathemtics with the Axion of Choice or without it.

Which way is better?

[Cabbage]

Sure, some mathematics can be developed without AC, but we also lose a lot without it. Without it, to name a few examples:

1. We don’t have infinite product spaces.
2. Not every vector space has a basis.
3. A given set may not have a cardinality.
4. All of the equivalents to AC, of course, such as Zorn’s Lemma, Well ordering theorem, Hausdorff maximality principle, and the Tychonoff theorem.

Some of the most interesting mathematics being researched today depends on the axiom of choice; I doubt there are any modern mathematicians of significance who reject AC.

[themoon]

Neither way is “better.” Both are internally consistent.

[ultrafilter]

Well, either it’s true, or it’s not. Does what I want even matter? ZFC and ZF~C (set theory with and without the axiom of choice) are both equally consistent, so they’re both “good” systems. I happen to like the axiom of choice, so I’m going to use it. And like Cabbage says, most mathematicians these days do accept choice.

I recall reading that there’s still some controversy over the axiom of foundation:

In set theory, the axiom of regularity, also known as the axiom of foundation, is that for every set S there is an element a in it which is disjoint from S. Under the axiom of choice, this axiom is equivalent to saying there is no infinite sequence {a[sub]n[/sub]} such that a[sub]i+1[/sub] is a member of a[sub]i[/sub]. Some corollaries are that no set belongs to itself, since otherwise {S} would violate the axiom of regularity.

Source: http://www.wikipedia.com/wiki/Axiom_of_regularity

[The_Ryan]

I’m rather bothered by the idea of taking a sphere and then putting it back together again into a larger sphere. Also, any proof that involves AC is by necessity nonconstructive. Just how useful is the statement “A set with these properties exists, but there’s absolutely no way that we can figure out what it is”?

[Cabbage]

Neither way is “better.” Both are internally consistent.

Well, the consistency of ZF set theory implies the consistency of ZF theory plus the axiom of choice. While I (and most other mathematicians) think ZF set theory is probably consistent, it can’t be proven that it’s consistent.
About the axiom of foundation, no mathematics I know of relies on it; it’s more of a “technical” axiom than anything else, and not really useful in any modern math. It basically restricts our universe of sets to the well-founded sets, where we do all of our mathematics, anyway. Plus, it gets rid of bizarre things like sets containing themselves.

The Ryan:

Of course, the catch there is that those pieces you divide the sphere into are so pathological that there’s no meaningful way to define a volume for them. It is surprising, but maybe not so surprising in light of that.

Also, I think the statement, “A set with these properties exists, but there’s absolutely no way that we can figure out what it is” is quite useful. There are lots of things, such as well orderings of sets, infinite products, and ultrafilters, which are extremely interesting mathematically, providing many fruitful areas of research, which simply cannot be constructed algorithmically.

[december]

*Originally posted by The Ryan *
**I’m rather bothered by the idea of taking a sphere and then putting it back together again into a larger sphere. Also, any proof that involves AC is by necessity nonconstructive. Just how useful is the statement “A set with these properties exists, but there’s absolutely no way that we can figure out what it is”? **

Somewhere around 1968, I took a class in which the Banach-Tarski theorem was proved. I vaguely recall that it used group theory as well as set theory and measure theory. I recall that the subsets are obviously not measurable sets. After 5 years of graduate math courses I finally had enough background to follow the proof. Whoopie!

My answer to the question of how useful the statment is would be, “It’s not useful at all.” As The Ryan implies, since it’s non-costructive, and since the subsets are not measurable, there’s no application to any kind of real-world situation.

Cabbage, I thank you for reminding me about the theorem of every vector space having a basis, which I had long since forgotten. In fact, I’m no longer sure how one defines a basis for an infinite-dimentional vector space. Would it be that each point can be represented by a unique finite linear combination of basis elements?

This vector space result is also non-constructive, I guess, so it would also not be useful for real-world problems. It’s elegant, though.

IMHO the A of C yields many elegant mathematical results, but lacks real-world applicability. Whether to use or not might depend on which of these things you’re seeking

[Cabbage]

In fact, I’m no longer sure how one defines a basis for an infinite-dimentional vector space. Would it be that each point can be represented by a unique finite linear combination of basis elements?

Yeah, that’s right.

IMHO the A of C yields many elegant mathematical results, but lacks real-world applicability. Whether to use or not might depend on which of these things you’re seeking.

That may or may not be true, I’m afraid I’m not familiar enough with applications of mathematics to be certain. I wouldn’t rule out the possibility of applications having been found (or that they will be found); for example, back with the vector space example, aren’t Hilbert spaces used extensively in quantum mechanics? Existence of bases there could be important. And, in general, it’s very easy for me to believe that certain mathematical objects that depend on AC for their “existence” may have many applications in the real world.

[Orbifold]

*Originally posted by Cabbage *
I wouldn’t rule out the possibility of applications having been found (or that they will be found); for example, back with the vector space example, aren’t Hilbert spaces used extensively in quantum mechanics?

Not to mention Fourier analysis; the Fourier transform is just a matter of expressing an element of a Hilbert Space in terms of a given basis.

[John_Kentzel-Griffin]

I wish I knew who said this, but I think this might amuse:

The Axiom of Choice is obviously true.
The Well Ordering Principle is obviously false.
And who can tell about Zorn’s Lemma?

[december]

*Originally posted by DrMatrix *
**I wish I knew who said this, but I think this might amuse:

The Axiom of Choice is obviously true.
The Well Ordering Principle is obviously false.
And who can tell about Zorn’s Lemma? **

Jerry Bona once said,

The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn’s Lemma?

This is a joke. In the setting of ordinary set theory, all three of those principles are mathematically equivalent – i.e., if we assume any one of those principles, we can use it to prove the other two. However, human intuition does not always follow what is mathematically correct. The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians; and Zorn’s Lemma is so complicated that most mathematicians are not able to form any intuitive opinion about it.

http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

[ultrafilter]

Q: What’s yellow and equivalent to the axiom of choice?
A: Zorn’s lemon.

So, anyone else know any good jokes about the axiom of choice?

[Cabbage]

What’s equivalent to the axiom of choice and jumps off cliffs?

[John_Kentzel-Griffin]

*Originally posted by ***ultrafilter **
Well, either it’s true, or it’s not. Does what I want even matter? ZFC and ZF~C (set theory with and without the axiom of choice) are both equally consistent, so they’re both “good” systems. I happen to like the axiom of choice, so I’m going to use it. And like Cabbage says, most mathematicians these days do accept choice.

I like the Axiom of Choice and, as has been pointed out, it is quite useful. But I want to comment on ultrafilter’s comment with a slight change to his notation. ZFC and ZF (set theory with and without AoC) are both equally consistent. Yes, but since AoC is independent, the negation is also independent. That is ZF~C (ZF assuming AoC is false) and ZF are equally consistent. I have trouble wrapping my mind around ZF~C – Set theory with at least one set that has no choice function, but it is as consistent as ZF. AoC is rather like the parallel postulate, Godel’s statement, and the Continuum Hypothesis in this regard.

But to return to the hijack:

*Originally posted by ***Cabbage **
What’s equivalent to the axiom of choice and jumps off cliffs?

I give. (Does it have anything to do with Tychonoff theorem?)

[ubergoober]

*Originally posted by Cabbage *

About the axiom of foundation, no mathematics I know of relies on it; it’s more of a “technical” axiom than anything else, and not really useful in any modern math. It basically restricts our universe of sets to the well-founded sets, where we do all of our mathematics, anyway. Plus, it gets rid of bizarre things like sets containing themselves.

**

There are some areas on the borders of mathematics where the axiom of foundation is annoying. It’d be nice to describe computer programs as sets of potential states a computer could be in. However, since some computer programs return to their intitial states, you’d better not describe them in terms of sets if you want the axiom of foundation.

As far as I can tell, the axiom of foundation is there to avoid getting in trouble with Russel’s paradox. It’s stricter than necessary, but it doesn’t really limit any mathematics. Weaker axioms that still avoid the paradox are of interest to computer scientists and philosophers with mathematical tendencies.

[Cabbage]

What’s equivalent to the axiom of choice and jumps off cliffs?

Zorn’s lemming!

(Yeah, I know lemmings don’t actually jump off cliffs)

[december]

What’s the equivalent of the A of C and requests water for his school?

[ultrafilter]

*Originally posted by ubergoober *
**There are some areas on the borders of mathematics where the axiom of foundation is annoying. It’d be nice to describe computer programs as sets of potential states a computer could be in. However, since some computer programs return to their intitial states, you’d better not describe them in terms of sets if you want the axiom of foundation.

As far as I can tell, the axiom of foundation is there to avoid getting in trouble with Russel’s paradox. It’s stricter than necessary, but it doesn’t really limit any mathematics. Weaker axioms that still avoid the paradox are of interest to computer scientists and philosophers with mathematical tendencies. **

That sounds interesting. Any recommendations for further reading?

[ubergoober]

*Originally posted by ultrafilter *
**

*Originally posted by ubergoober *
**
As far as I can tell, the axiom of foundation is there to avoid getting in trouble with Russel’s paradox. It’s stricter than necessary, but it doesn’t really limit any mathematics. Weaker axioms that still avoid the paradox are of interest to computer scientists and philosophers with mathematical tendencies. **

That sounds interesting. Any recommendations for further reading? **

I’d recommend the book Viscious Circles by John Barwise and Larry Moss.