Axiom of choice, Banach-Tarski and reality

Question

The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site.

One of the non-obvious consequences of the axiom of choice is the Banach-Tarski paradox and thus the existence of non-measurable sets.

On the other hand, there seem to be models of Zermelo-Fraenkel set theory without axiom of choice where every set would be measurable.

What does this say about the "plausibility" of the axiom of choice? Are there reasons why it is plausible (for physicists, philosophers, mathematicians) to believe that not all sets should be measurable? Is the Banach-Tarski paradox one more reason why one should "believe" in the axiom of choice, or is it on the opposite shedding doubt on it?

Answer 1

There are two ingredients in the Banach-Tarski decomposition theorem:

1. The notion of space, together with derived notions of part and decomposition.
2. The axiom of choice.

Most discussion about the theorem revolve around the axiom of choice. I would like to point out that the notion of space can be put under scrutiny as well.

The Banach-Tarski decomposition of the sphere produces non-measurable parts of the sphere. If we restrict the notion of "part" to "measurable subset" the theorem disappears. For instance, if we move over into a model of set theory (without choice) in which all sets are measurable, we will have no Banach-Tarski. This is all well known.

Somewhat amazingly, we can make the Banach-Tarski decomposition go away by extending the notion of subspace, and keep choice too. Alex Simpson in Measure, Randomness and Sublocales (Annals of Pure and Applied Logic, 163(11), pp. 1642-1659, 2012) shows that this is achieved by generalizing the notion of topological space to that of locale. He explains it thus:

"The different pieces in the partitions defined by Vitali and by Banach and Tarski are deeply intertangled with each other. According to our notion of “part”, two such intertangled pieces are not disjoint from each other, so additivity does not apply. An intuitive explanation for the failure of disjointness is that, although two such pieces share no point in common, they nevertheless overlap on the topological “glue” that bonds neighbouring points in $\mathbb{R}^n$ together."

Peter Johnstone explained in The point of pointless topology why locales have mathematical significance that goes far beyond fixing a strange theorem about decomposition of the sphere. Why isn't everyone using locales? I do not know, I think it is purely a historic accident. At some point in the 20th century mathematicians collectively lost the idea that there is more to space than just its points.

I personally prefer to blame the trouble on the notion of space, rather than the axiom of choice. As far as possible, geometric problems should be the business of geometry, not logic or set theory. Mathematicians are used to operating with various kinds of spaces (in geometry, in analysis, in topology, in algebraic geometry, in computation, etc.) and so it seems only natural that one should worry about using the correct notion of space first, and about underlying foundational principles later. Good math is immune to changes in foundations.

Answer 2

It's notable that most of the "bread and butter" mathematical consequences of the axiom of choice are actually consequences of countable choice. (Every infinite set contains a countable subset, a countable union of countable sets is countable, etc.) The Hahn-Banach theorem is a counterexample, but only if you want it for nonseparable spaces, and I can't think of any time I've ever needed this. When restricted to separable Banach spaces it doesn't require any choice principle at all! Whereas the seemingly pathological consequences of choice (existence of nonmeasurable sets, Banach-Tarski, well-ordering of the real line) generally do not follow from countable choice.

So the argument from mathematical value seems to me to support countable choice more than full choice. But that isn't a very strong argument, is it? We can't decide whether an axiom is true based on whether we like its consequences. At best it's suggestive.

Incidentally, I had the impression when I read Zermelo that he had great polemical skill, but none of his arguments seemed to get directly to the truth of the axiom. He argues for the mathematical value of the axiom. He points out that his critics have themselves on occasion unwittingly used the axiom, which is a devastating point, but has little bearing on the question of truth. (If I'm not mistaken, those unwitting uses were all of countable choice, by the way.)

You ask if one should "believe" the axiom of choice, and I think you are right to put the word "believe" in quotes. I feel strongly that set-theoretic assertions are objectively meaningful, but I also feel that philosophers of mathematics have done a very poor job of clarifying what sets are. (Halmos: "A pack of wolves, a bunch of grapes, or a flock of pigeons are all examples of sets of things." Black: "It ought then to make sense, at least sometimes, to speak of being pursued by a set, or eating a set, or putting a set to flight.") If we can't even get that straight, it's hard to come to grips with questions about the truth of questionable axioms.

Answer 3

The other answers don't seem to have said much about why the axiom of choice is widely regarded as plausible. Let me try to address that question.

First let's dispose of some non-reasons. In response to your questions, I don't know of anyone who thinks that the Banach–Tarski paradox is a reason to believe in the axiom of choice. I also don't know of anyone who argues, "It is a priori plausible that there exist non-measurable sets; so the fact that the axiom of choice yields the attractive conclusion that there are non-measurable sets is a point in favor of believing the axiom of choice." Instead, those who are comfortable with the existence of non-measurable sets typically start by accepting the axiom of choice, and then they accept non-measurable sets as "part of the territory" that comes with the axiom of choice.

Those who think that Banach–Tarski casts doubt on the axiom of choice typically have a philosophical predisposition that math is supposed to model the physical world closely. So for example, $\mathbb R^3$ is not just a random mathematical structure that we study purely for its own sake; it is supposed to be a decent model of physical space (or at least, open subsets of $\mathbb R^3$ are supposed to model localized regions of physical space). Banach–Tarski, when given a direct physical interpretation in this way, yields something that we "know" makes no physical sense, and so if we think that math is supposed to yield physical truth in this way, then Banach–Tarski is going to lead us to reject something in the math. Whether that "something" we reject is the axiom of choice is a separate question, and Andrej Bauer's excellent answer shows that there are other options, but the point I want to highlight is that we're going to be led down this path in the first place only if we have certain presuppositions about how math and physics are supposed to relate.

There are others who don't view set theory in this way. According to them, set theory is supposed to be about abstract collections of things, and the way to arrive at axioms is by abstractly thinking about what properties they should have, not by comparing them with the physical world. The axiom of choice can be thought of as saying that if you have a bunch of nonempty collections of things, then there is another collection of things that contains one element from each of your original nonempty collections. Stated this way, the principle sounds intuitively plausible, and I would argue that this intuitive plausibility is, at least implicitly, the main argument in the minds of most people who accept the axiom of choice. If this is the way you think, then non-measurable sets and Banach–Tarski are not going to dissuade you from accepting the axiom of choice. Those phenomena will just lead you to say that we can't arrive at physical predictions from mathematics in such a naive manner; instead, to do physics, we have to formulate physical theories. Math can of course help a lot with the construction of physical theories, but it's not as simple as just saying that the mathematical theory of $\mathbb R^3$ is our theory of physical space.

These two options aren't the only options. The work of Solovay shows that you can, to a large extent, have your cake and eat it too, by working in a set-theoretic universe where all subsets of $\mathbb R^n$ are Lebesgue-measurable and a weakened, but still quite strong, version of the axiom of choice known as "dependent choice" is available. Why Solovay's model hasn't become more popular is not completely clear, but perhaps part of the reason is that it feels like a "compromise position," and the people in the two different camps above have not seen any need to migrate to that kind of compromise.

Answer 4

Are there reasons why it is plausible (for physicists, philosophers, mathematicians) to believe that not all sets should be measurable?

Yes. If every set of reals is Lebesgue measurable, then you can partition $\mathbb{R}$ into more than continuum many pairwise disjoint non-empty pieces. (See this answer and comments for the details.)

Surely the Banach-Tarski paradox seems unintuitive. But having a set that can be broken up into more pieces than there originally were... is just wrong.

Answer 5

Arguments from physics may not help. Here is Bryce DeWitt reviewing Stephen Hawking and G.F.R. Ellis using the axiom of choice in 1973:

The book also contains one failure to distinguish between mathematics and physics that is actually serious. This is in the proof of the main theorem of chapter 7, that given a set of Cauchy data on a smooth spacelike hypersurface there exists a unique maximal development therefrom of Einstein’s empty-space equations. The proof, essentially due to Choquet-Bruhat and Geroch, makes use of the axiom of choice, in the guise of Zorn’s lemma. Now mathematicians may use this axiom if they wish, but it has no place in physics. Physicists are already stretching things, from an operational standpoint, in using the axiom of infinity.

It is not a question here of resurrecting an old and out-of-date mathematical controversy. The simple fact is that the axiom of choice never is really needed except when dealing with sets and relations in non-constructive ways. Many remarkable and beautiful theorems can be proved only with its aid. But its irrelevance to physics should be evident from the fact that its denial, as Paul Cohen has shown us, is equally consistent with the other axioms of set theory. And these other axioms suffice for the constructions of the real numbers, Hilbert spaces, C* algebras, and pseudo-Riemannian manifolds–that is, of all the paraphernalia of theoretical physics.

In “proving” the global Cauchy development theorem with the aid of Zorn’s lemma what one is actually doing is assuming that a “choice function” exists for every set of developments extending a given Cauchy development. This, of course, is begging the question. The physicist’s job is not done until he can show, by an explicit algorithm or construction, how one could in principle always select a member from every such set of developments. Failing that he has proved nothing.

Some physicists want to use the axiom of choice, but some physicists don't.

Answer 6

Just a quick addendum: the result was actually found by Felix Hausdorff and then repackaged in a more spectacular form by Banach and Tarski. Hausdorff' point was precisely to show that the axiom of choice leads to such unreasonable consequences that it should probably be avoided. This was a hot topic of discussion during the following decades, and it seems that the answer at large from the mathematical community is: ok, there are inconveniences, but the advantages of using the AC are superior and we prefer to have it available.

Answer 7

I think one of the strongest arguments for the axiom of choice is that every model of ZF contains as an inner model a constructible universe $L$, and $AC$ is a theorem of the constructible universe. We have

$$ ZF+(V=L) \vdash AC $$

In other words, a necessary condition for asserting $\neg AC$ is to first assert "there exists a set that cannot be constructed" — that is, it requires positing the existence of additional structure above and beyond what is guaranteed by ZF.

Thus, it seems clear to me that ZFC is by far the better choice for foundations. One may still wish to work with another set theoretic universe, but that's most appropriately done as the study of additional structure built atop of the foundations, not by rewriting the foundations themselves.

Answer 8

Physical applications of the Banach–Tarski theorem were explored by Henry Kuttner, The Time Axis, Startling Stories 18:3 (January 1949), 13–82. Some excerpts from pp. 66–67:

"'Professor Raphael M. Robinson of the University of California now shows that it is possible to divide a solid sphere into a minimum of five pieces and reassemble them to form two spheres of the same size as the original one. Two of the pieces are used to form one of the new spheres and three to form the other.

"'Some of the pieces must necessarily be of such complicated structure that it is impossible to assign volume to them. Otherwise the sum of the volumes of the five pieces would have to be equal both to the volume of the original sphere and to the sum of the volumes of the two new spheres, which is twice as great.'"

[. . . .]

"This is it," he said.

Even the crowd around the neural-web table thinned as the workers in the laboratory flocked around him to watch.

He had a sphere about the size of a grapefruit, floating in mid-air above his table. He did things to it with quick flashes of light that acted exactly like knives, in that it fell apart wherever the lights touched, but I got the impression that those divisions were much less simple than knife-cuts would be. The light shivered as it slashed and the cuts must have been very complex, dividing molecules with a selective precision beyond my powers of comprehension.

The sphere floated apart. It changed shape under the lights. I am pretty sure it changed shape in four dimensions, because after a while I literally could not watch any more. The shape did agonizing things to my eyes when I tried to focus on it.

When I heard a long sigh go up simultaneously from the watchers I risked a look again.

There were two spheres floating where one had floated before.

Answer 9

I think it is interesting to re-phrase this question relative to other axioms and/or theorems of ZFC. What Andrej Bauer's answer suggests is that it may not be the axiom of choice per se that is the culprit, but rather the underlying structure.

For example, it is provable that the existence of non-Lebesgue measurable sets and the Banach–Tarski paradox are both an implication over ZF (without Choice) of the Hahn–Banach theorem (HB) in functional analysis. This means that we can prove those "pathologies" as a theorems from ZF+HB.

Another way to put it is to analyze the structure within the context of different orders of logic. In ordinary first-order logic Choice is provably equivalent to the well-ordering theorem (WO) over ZF. What is different in second-order logic is that WO is strictly stronger than choice: WO $\vdash_{ZF}$ Choice, but Choice $\nvdash_{ZF}$ WO.

In other words, one may get the feeling that perhaps there has been too much emphasis put on Choice. Other axioms and theorems can prove just as problematic. The approach described above is the essence of Reverse Mathematics (RM). While RM has been traditionally carried out at a much lower proof-theoretic strength level (subsystems of second-order arithmetic), in my opinion this provides a very useful framework for analyzing the foundations of other parts of mathematics.