Undercurrent

That's just like my job (job title is "mathematician", but do an inordinate amount of coding. )

m.

CardinalMan

Mmmmm... set theory.

I think that wade-w's description of the Axiom of Choice above is wonderful and accurate, but if you're interested in reading a little more, I've written a (currently incomplete) webpage entitled <a href="http://www.math.uiuc.edu/~mileti/choice.html" target="_blank"> The Axiom of Choice and Zorn's Lemma</a>. Maybe this discussion will give me the motivation to finish it.

Since it has not yet been mentioned, there is a theorem of mathematical logic that says that the Axiom of Choice is independent of the other axioms of set theory. In other words, if the other axioms of set theory are consistent (i.e. that you can not derive a contradiction from them), then it follows that you will not create an inconsistency by adding the Axiom of Choice, and also you will not create an inconsistency by adding the negation of the Axiom of Choice. Hence, if you accept the other axioms of set theory, there is no purely mathematical reason for either accepting or rejecting the Axiom of Choice. Your reasons for its acceptance or rejection must come from another perspective (philosophical, theological, aesthetic, pragmatic, experimental, etc.).

Although some mathematicians claim that they accept the Axiom of Choice because it is known to be consistent with the other axioms, I believe that most accept it simply because so much modern mathematics relies on it in a provably essential way, and hence its rejection would involve eliminating a huge amount of the beautiful mathematics that was developed in the twentieth century. A couple of very important examples were mentioned above (the fact that every vector space has a basis, and the fact that every proper ideal in a ring with identity can be extended to a maximal ideal). Both of these are crucial facts that are used time and time again, and without them, the fields of Linear Algebra and Commutative Algebra lose not only power, but also a great deal of elegance. Here are a few other important theorems are mathematics that are known to be unprovable without the Axiom of Choice:

The equivalence between various definitions of limits and continuity (the standard epsilon-delta definition versus definition via convergent sequences). This equivalence is fundamental to modern analysis (also this can proved via a countable version of the Axiom of Choice).

The Tychonoff Product Theorem: This theorem is known to be equivalent to the Axiom of Choice if you assume the other axioms of set theory. It says that the arbitrary product of compact topological spaces is compact. This theorem is crucial to certain branches of topology, analysis, and descriptive set theory.

The Hahn-Banach Thorem: This is one of the fundamental theorms of functional analysis.

Existence of Nonprincipal Ultrafilters: This result is used constantly in fields like model theory and nonstandard analysis.


I accept the Axiom of Choice because I am unwilling to discard so many beautiful parts of mathematics simply due to philosophical trepidation. Of course, we have no physical experience with infinite collections of arbitrary sets, so we have no "scientific" reasons to accept it. However, as I say in the webpage above:

Without the Axiom of Choice, mathematicians are practically helpless when dealing with infinity. There are a huge number of extremely basic questions in mathematics that can not be decided if we reject it. In fact, there still are a large number of basic questions in mathematics that can not be decided even with the Axiom of Choice.

As a result, my view of mathematics is as follows. Although mathematics originated to reflect the essential abstract parts of physical reality, there is no longer any reason why it should play a subserviant role to physical reality. No amount of scientific experimentation or detached philosophical thought will settle the question "Is the Axiom of Choice true?". Mathematics has entered an age in which it has its own questions, its own methodology, and its own reasons for existence. In my opinion, the Axiom of Choice furthers the practice of mathematics, a field of study that searches for structure in the realm of number and shape.

In fact, my philosophy of mathematics extends much further. When we discover provably unsolvable problems in mathematics (which we will perennially do due to Goedel's Incompleteness Theorem) that can not be answered by appeal to our senses, we should proceed in a manner that advances mathematics for the sake of mathematics. We should reach conclusions based on mathematical analogies and the amount of elegance and beauty that will be obtained by accepting the new axioms. I also think that we should strive to make mathematics as general and all-encompassing as possible. Therefore, I oppose adopting the axiom V = L (a statement which implies the Axiom of Choice and decides many independent questions) as it constricts mathematics to the bare minimum set-theoretic universe. Personally, I am quite fond of adopting various large cardinal axioms (which assert the existence of infinite sets of ridiculously large size) because they expand the mathematical universe and they answer many independent questions in a manner that is both natural and aesthetically pleasing.

CardinalMan

[Edited to add an important negation]

[ October 27, 2002: Message edited by: CardinalMan ]</p>

godlessmath

Excellent treatise Cardinal Man. Perhaps there is hope for the AC yet. Now I'm wondering, if there are non-trivial results from assuming the axioms of set theory and the negation of AC? I know of none personally. I wonder if it creates as interesting mathematical worlds as non-Euclidean geometry does.

Goliath

Well said, CardinalMan!

I, too, wonder if there has been anything interesting that has developed using the negation of the axiom of choice.

However, being a commutative ring theorist, a very important theorem in my field depends on Zorn's lemma: the fact that any proper ideal of a commutative ring with identity is contained inside of a maximal ideal.

This doesn't really trouble me, however, since the axiom of choice is independent of the remaining axioms of mathematics, and that it seems in many ways to "fit in" (in a very loose, touchy-feely way) with the rest of mathematics. Mathematics is hard enough with the axiom of choice. Even thoug the axiom of choice has some goofy consequences (ie there exists a well-ordering of every set), I couldn't imagine how much harder things would be without it.

Sincerely,

Goliath

CardinalMan

That's a very interesting question godlessmath. I don't think that you can get very much information simply by assuming the negation of the Axiom of Choice. While the Axiom of Choice asserts a kind of uniform regularity of sets, its negation simply says that there exists at least one "pathological" example of a collection of nonempty sets without a choice function. As far as I can tell, it gives no information about what this set might be.

However, there is at least one, sometimes appealing, axiom called that Axiom of Determinacy which does contradict the full Axiom of Choice. Before I can state it, I need to talk about games.

Let X be an arbitrary set of infinite sequences of natural numbers. Play the following game with two players named I and II. At the first stage, player I chooses a natural number a_1. At the second stage, player II chooses a natural number a_2. Continue in this way alternating turns so that player I chooses a_n at odd stages n and player II chooses a_n at even stages n. After all stages are complete, an infinite sequence of natural numbers a_1,a_2,a_3,... is produced. If this sequence lies in our set X, then player I wins, and if it does not lie in X, then player II wins.

Call the game based on the set X determined if some player has a winning strategy (i.e. has a way to decide his or her moves so that, regardless of the moves of the other player, he or she will be able to ensure a win). The Axiom of Determinacy says that the game based on X is determined for every set of infinite sequences of natural numbers.

At first glance, the Axiom of Determinacy seems to be quite pleasing. However, one can show, without too much difficulty, that it contradicts the full Axiom of Choice. On the other hand, the Axiom of Determinacy, together with a fairly weak form of the Axiom of Choice called the Axiom of Dependent Choice, implies some extremely pleasing facts in analysis. For example, the following statements, while refutable from the full Axiom of Choice, are provable from ordinary axioms of set theory together with the Axiom of Determinacy and the Axiom of Dependent Choice:

1) Every set of real numbers is Lebesgue measurable.

2) Every set of real numbers has the Baire property (i.e. for every set of real numbers A, there exists an open set of real numbers U, such that the symmetric difference of A and U is meager; in other words every set of real numbers is almost an open set of real numbers).

3) Every uncountable set of real numbers has a perfect subset.

So why don't mathematicians prefer the Axiom of Determinacy to the Axiom of Choice? While the Axiom of Determinacy has very nice consequences in analysis, it does have some rather strange consequences elsewhere. The set-theoretic universe looks quite bizarre under the Axiom of Determinacy, and there are some strange consequences in my own field of computability theory. Furthermore, most mathematicians with an interest in these questions seem to believe that arbitrary subsets (of reals or of infinite sequences of natural numbers) simply shouldn't be as "nice" as the Axiom of Determinacy forces them to be. In other words, a completely arbitrary subset should be able to have a certain richness that is ruled out under the Axiom of Determinacy.

In fact, the whole field of descriptive set theory tries to understand arbitrary subsets of reals (or more general so-called Polish spaces like the collection of infinite sequences of natural numbers) via hierarchies that capture just how "definable" they are. You're probably familiar with the Borel Hierarchy of sets of real numbers. Well, it turns out that if the set X above is a Borel set, then one can prove in set theory with the regular Axiom of Choice that the game based on X is determined. Thus, even under the Axiom of Choice where certain subsets can behave "pathologically" in these games, every "nicely defined" collection of infinite sequences of natural numbers behaves well.

Furthermore, there is a hierarchy of sets called the Projective Hierarchy which extends the Borel hierarchy to more sets, which have less structure than Borel sets, but are still "definable" in a certain sense. It seems that the Axiom of Choice does not contradict Projective Determinacy, the statement that for any projective set X, the game based on X in determined. However, the Axiom of Choice does not imply projective determinacy either. In fact, the study of Projective Determinacy is closely tied to the study of large cardinals that I alluded to at the end of my previous post.

What a wonderful time to be a mathematician.

CardinalMan

[Edited for spelling]

[ October 27, 2002: Message edited by: CardinalMan ]</p>

godlessmath

This makes me drool with delight. Non-standard analysis is something that I'd like to see before I check out. I've only seen a bit about hyperreals, I know how you get them, but I've never seen them in action.

When, oh when, has there ever been a time when it wasn't wonderful to be a mathematician?

This is all too interesting and I have to go to bed. It's a cruel and unfair world!

Clutch

Heya, mathies,

Great thread!

A question, though: How much of the "beautiful math" that relies upon the industrial-strength AC is known to be unrecoverable constructively? (I guess I mean, known by constructive standards?)

godlessmath

No clue, but I have a feeling what some may be, like that infamously non-lebesgue measurable set.

All of this stuff reminds me too much of Euclid and the parallel postulates. Perhaps there is an isomorphism between the two.

CardinalMan

You must want to be an analyst. Every construction of the hyperreals that I've encountered used the existence of nonprincipal ultrafilters, which essentially relies on the Axiom of Choice.

There were times as a undergraduate when I felt that every interesting aspect of mathematics had already been solved, and all that remained was minutiae. I've since learned how naive that belief was.

CardinalMan

[ October 28, 2002: Message edited by: CardinalMan ]</p>

CardinalMan

Hmmm... I'm not sure I fully understand your question. I suppose that it depends on the version of constructivism you have in mind. If we formalize set theory using a first-order language, then all of these implications can be proved finitistically (and hence "constructively") in this system, just like in any other formal system. However, a constructivist would certainly interpret this formal system differently than a "standard" mathematician.

Many constructivists reject concepts like an arbitrary set and the completed infinity (and thus reject many other aspects of set theory), so it would seem to me that the Axiom of Choice and many of the above statements would be meaningless to them.

I'm certainly not an expert in constructivist philosophy of mathematics, so I may very well be misunderstanding your question completety, Clutch. Let me know if I'm way off the mark.

CardinalMan