How can you tell when your theory has overstepped the bounds of reasonableness? How about when you start telling people your “facts” and their faces register with incredulity and disbelief? That is the response of most reasonable people when they hear about the “Banach-Tarski paradox”.

From Wikipedia:

The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembly.

The “theorem” is commonly phrased in terms of two solid balls, one twice the radius of the other, in which case it asserts that we can subdivide the smaller ball into a small number (usually 5) of disjoint subsets, perform rigid motions (combinations of translations and rotations) to these sets, and obtain a partition of the larger ball. Or a couple of balls the same size as the original. It is to be emphasized that these are cut and paste congruences! This was first stated by S. Banach and A. Tarski in 1924, building on earlier work of Vitali and Hausdorff.

This “theorem” contradicts common sense. In real life we know that it is not easy to get something from nothing. We cannot take one dollar, subtly rearrange it in some clever fashion, and end up with two dollars. It doesn’t work.

That is why most ordinary people, when they hear about this kind of result, are at first disbelieving, and then, when told that the “proof” involves “free groups of rotations” and the “Axiom of Choice”, and that the resulting sets are in fact impossible to write down explicitly, just shake their heads. Those pure mathematicians: boy they are smart, but what arcane things they get up to!

This theorem is highly dubious. It really ought to be taken with a grain of salt, or at least generate some controversy. This kind of logical legerdemain probably should not go unchallenged for decades.

The logical flaws involved in the usual argument are actually quite numerous. First there are confusions about what “free groups” are and how we specify them. The definition of a finite group and the definition of an “infinite group” are vastly different kettles of fish. An underlying theory of infinite sets is assumed, but as usual a coherent theory of such infinite sets is missing.

Then there is a claim that free groups can be found inside the group of rotations of three dimensional space. This usually involves some discussion involving real numbers and irrational rotations. All the usual difficulties with real numbers that students of my YouTube series MathFoundations will be familiar with immediately bear down.

And then finally there is an appeal to the *Axiom of Choice*, from the ZFC axiomfest, which claims that one can make an infinite number of independent choices. But this contradicts the *Law of (Logical) Honesty* that I put forward several days ago. I remind you that this was the idea:

Don’t pretend that you can do something that you can’t.

You cannot make an infinite number of independent choices. Cannot. Impossible. Never could. Never will be able to. No amount of practice will help. Whistling while you do it won’t make it happen. You cannot make an infinite number of independent choices.

So we ought not to pretend that we can; that is what the Law of (Logical) Honesty asserts. We can’t just say: and now let’s suppose that we can make an infinite number of independent choices. That is just an empty phrase if we cannot support it in ways that people can observe and validate.

The actual “sets” involved in the case of transforming a ball of radius 1 to a ball of radius 2 are not sets that one can write down in any meaningful way. They exist only in a kind of no-mans land of speculative thinking, entirely dependent on these set-theoretic assumptions that pin them up. Ask for a concrete example, and explicit specifications, and you only get smiles and shrugs.

And so the Banach-Tarski nonsense has no practical application. There is no corresponding finite version that helps us do anything useful, at least none that I know of. It is something like a modern mathematical fairy tale.

Shouldn’t we be discussing this kind of thing more vigorously, here in pure mathematics?

EmThere isn’t really worth in attacking AC (the Axiom of Choice) over the other ZF axioms. This is because if ZF is consistent then so is ZFC (that is, ZF + AC). This was shown by Gödel. In fact, he did the following: Working only from the ZF axioms (no Axiom of Choice) he identified the “constructible universe” L, which is a certain specific collection of sets (L is not a set itself, as it is too large), and he proved that inside L, all axioms of ZFC are true, and actually, there is a definable (and very natural) class well order of all sets. (The sets in L are all “constructed”, and the class well order is basically the order in which they are constructed. The construction takes place along the ordinals.) It follows that in L, AC is not only true (hence also the Banach-Tarski Paraodx), but AC is true in a very concrete manner. For example, working inside L, given any set A of non-empty sets, there is not only a choice function for A, but there is a natural “least” such choice function for A, and this gives it a concrete definition from the parameter A. (The definition needs to be interpreted inside L.)

njwildberger: tangential thoughtsPost authorEm: You are swallowing the standard cliches without even acknowledging that their meaning is questionable. “Axioms” are here being bandied about without any consideration of whether or not they actually correspond to intuitively obvious facts—-that is the original, and correct, usage of the term. To give but one example; you refer to Gödel’s “constructible universe” L as “a certain specific collections of sets”. I would like you to show me L. Because I suspect L is about as real as the astral plane of Leprechaun dreamings.

EmDo you have an opinion about whether ZF is consistent or not? I don’t claim that there is some one correct universe of sets in some platonic realm. I am basically just interested in what is provable from ZF and variants thereof. So when I say “a certain specific collection of sets”, I just mean relative to a given model of ZF. But assuming that ZF is consistent (hence so is ZFC+V=L, where V denotes the full set theoretic universe, and V=L asserts that every set is constructible), then it is not too difficult to construct pretty explicitly a model of ZFC+V=L. One does this through a variant of the proof of completeness/compactness for first order logic. In fact the model, with its membership relation, is definable over the naturals N (where N comes equipped with the usual +, X, <). (So from our external perspective, the model is countable, but the model doesn't see the enumeration of its elements that we see, so this is no contradiction.)

Here is a sketch of the construction: We first computably enumerate all of the sentences phi in the language of set theory, as _{n in N}. We then want to build a complete consistent theory T extending ZFC+V=L. (That is, T is consistent, and for every sentence phi, T proves either phi or its negation.) We do this recursively, deciding whether to put varphi_n or its negation into T at stage n. Because the theory produced by stage n is consistent, at least one of phi_n or its negation can be adjoined, maintaining consistency into stage n+1. In order to know how to navigate this, we need to know what things are provable from ZFC+V=L (thus, the process is not computable, but it is computable relative to an oracle for the halting problem, and thus is definable over N, with formulas coded by integers as usual).

Now consider all of the “sets” that T asserts exist and are uniquely definable from some formula in the language of set theory. Formally here, I mean to consider all of the *formulas* phi(v)

such that T has the sentence “there is a unique v such that psi(v)”.

We construct a model M whose elements are the equivalence classes of these formulas mod a certain equivalence relation, and define a notion of set membership for this model. For the equivalence relation, formulas psi(v), phi(v) are equivalent just when T includes the sentence

“there are v,w such that psi(v) and phi(w) and v=w”.

This is an equivalence relation because T is complete and consistent. (Transitivity: suppose psi equivalent with psi’ equivalent with psi”. Then T includes the sentences

(i) there is a unique v such that psi(v),

(ii) there is a unique v’ such that psi'(v’),

(iii) there is a unique v” such that psi”(v”),

(iv) there are v,v’ such that psi(v) and psi'(v’) and v=v’,

(v) there are v’,v” such that psi(v’) and psi”(v”) and v’=v”.

By completeness/consistency, T includes the conjunction of (i)–(v). But then it is straightforward to see that psi’ is equivalent with psi”.) We then define “membership” over the equivalence classes by:

[psi] “in” [phi]

iff

T includes the sentence “there are v,w such that psi(v) and phi(w) and v is in w”.

One now shows that this “membership” respects the equivalence relation, and that the structure produced satisfies all sentences in T, hence is a model of ZFC+V=L. A key fact needed here is that ZFC+V=L proves that there is a proper class well ordering <_L of the universe, which means that if T includes a statement of the form “exists x such that psi(x)'', then it also includes the sentence “there exists a unique x such that [psi(x) and for all y <_L x, not psi(y)]''. Thus, we get an element A in our model M from this statement, and one can show that that in our model, psi(A) is true, and thus M satisfies the statement “exists x psi(x)''.

The entire construction is computable relative to an oracle which tells you the halting problem, so not quite computable, but close to it. One might just say that the construction is completely explicit, but it involves a couple of quantifiers ranging over N, which I suppose you might not like. Of course by Gödel's incompleteness, the theory T above is not computable.