Many years ago, when I was an undergrad, I stumbled upon the Banach-Tarski Paradox (BTP), aka “one can cut a sphere into two disjoint parts which both have the same volume than the initial sphere“, and was quite puzzled by it — all the more since none of the courses I attended back then discussed it.
This is typically the kind of statement that, if not explained carefully, can drive people away from math (and sadly it probably has done so many times already). And of course it provides the perfect occasion to point out differences between physics and mathematics, as well as modifications of the undergraduate curriculum, so I’d like to discuss this a little bit. Comments welcomed (especially on the issue whether or not the countable version of the Axiom of Choice (AC) is enough to prove the BTP).
The wikipedia page about the BTP is well written, while another useful source is a discussion in the sci.math FAQ. Beside that, one should study a complete proof of the BTP, e.g. Terence Tao’s short story on the subject (he also has another very weak version of it here, but I won’t talk about it since it’s less surprising).
Of course, as mentionned in those documents, one can certainly think about many different possibilities for each of the words “cut”, “piece” and “reassemble”. As here one works with arbitrary subsets of as pieces, and with rotations and translations only (which are isometries, and in particular are measure-preserving) as means of reassembly, it’s clear that the trick will come from non-measurable sets.
As can be seen from Tao’s short story, the AC is what leads to the BTP, so a natural question is whether this can still be done if we use only the countable version of AC (a version which is accepted both by classical and constructivist mathematicians).
Now, it seems to me that it is the case in what Tao writes: AC is used to pick an element in each orbit of a -action on (where is a countable subgroup of , and some countable subset of the 2-sphere ), so the number of orbits — and hence of choices of elements — is countable since is so.
And yet, one reads in a paper by Foreman and Dougherty (published in PNAS in 1992) that Solovay has proved that within “ZF+countable AC” it is consistent to say that all subsets of are Lebesgue measurable, this under the assumption that “ZFC+there exists an inaccessible cardinal” is consistent. Their conclusion is then that the countable version of AC is not enough to construct non-measurable sets, and in particular it is not enough to prove the BTP.
But given the proof in Tao’s note it seems to me that countable AC is enough for proving the BTP, an so for producing non-measurable sets. Thus Solovay’s assumption of the consistency of “ZFC+there exists an inaccessible cardinal” is incorrect. Do you agree or have I overlooked something?
In that paper by Foreman and Dougherty, although I have not checked it, they also claim to show that a version of the BTP can be proved which does not require AC at all (countable or not), at the expense of working with open subsets of only, not arbitrary ones.
From this last result, as well as the wikipedia page mentionned earlier, it seems to me that the source of the paradox is not AC but in fact the very use of infinite sets, namely in the part where one considers the free group with two generators . Indeed the key fact there is that one can consider the three decompositions of into disjoint subsets , and this assumes only that such infinite sets are valid objects for one to manipulate.
The previous reasoning seems to be known, since in his paper Mesures finiement additives et paradoxes Pierre de la Harpe writes at the beginning “the mathematical paradoxes to be discussed here stem from our difficulty to think about infinity, and are recurring again and again“. The book by Stan Wagon on the BTP seems very complete, I’ll try to read it too.
It would be very nice to have a module quite early in the undergraduate mathematics curriculum solely devoted to the topic of infinity (yet another future project of mine is to one day write a book about that precisely fro undergraduates, with lots of examples, counter-examples, figures, epistemological discussions, …). The key thing to understand being the difference between a potential infinity (i.e. when things can in principle be done as many times as one wishes but of course without enumerating them all) and actual infinity (e.g. when one considers the set as an available object here and now), in particular the “absorbing” property of actual infinity (as witnessed for example in that fact that there is a bijection between and ).
That would make it easier for students to understand why our ZF(C)-based math is simply not a setting to allow to model all aspects of the physical world at once. And to see that physics really works the other way round, by going from facts to assumptions on the properties of the underlying objects, rather than from axioms (assumptions) to theorems (facts)…