The Banach-Tarski paradox: is it nonsense?

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.

Doubling_of_a_sphere,_as_per_the_Banach-Tarski_Theorem (1)

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?



20 thoughts on “The Banach-Tarski paradox: is it nonsense?

  1. Brandon

    I have come to find that there are two types of people in the world.

    There are those who, when confronted with the paradox, proclaim “This makes no sense at all! Let me search through the proof to find the error responsible for this horrendous logical inconsistency.”

    And there are those who, when confronted with the paradox, proclaim “This is amazing! The beauty of mathematics Transcends the comprehension of humankind! We are all but mere mortals.”

  2. Brandon

    I have come to find that there are two types of people in the world.

    There are those who, when confronted with paradoxes like this, proclaim “This makes no sense at all! Let me search through the proof to find the error responsible for this logical inconsistency.”

    And there are those who, when confronted with the paradox, proclaim “This is amazing! The beauty of mathematics Transcends my comprehension!”

  3. Tom W.

    Much that I love the idea of honesty, by that ‘law’, one should not then be able to say 0/n (zero divided by /any number) or 0 – n for that matter… and then n/0 shouldn’t be ‘undefined’ but on the contrary, well-defined as “n” since anything divided by nothing is anything ‘not-divided’ (i.e. undivided, whole), itself un-transformed… to overhaul mathematics with ‘honesty’ means a whole helluva lot more work than just dismissing much of the more ‘abstract’ theories out there… don’t you think?

  4. William

    I remember being super confused when my friend told me about this and I couldn’t believe it so I was trying to clarify that it wasn’t a metrical result, and was instead just about the congruence of pieces, so that the radius’ were just being labeled arbitrarily… Then I looked it up and found it was actually meant to be a result about euclidean space and was so bummed because I want to study analysis and I know the axiom of choice is goofy but I try and not think about it too much and hope it not THAT goofy… but you’re right. There are so many corollaries like this that are just ridiculous that sometimes i think i might have to switch specialties. I just don’t know what going on in a subject where the banach tarski paradx is true.

  5. Dan

    “That is just an empty phrase if we cannot support it in ways that people can observe and validate.”
    That made me think of the “sum” 1+2+3+4… =-1/12
    This can be observed by testing the strength of the Casimir effect, so does that make it true?

  6. Elias Torres

    Exactly, the first time I saw it I thought the same thing, when you cancel out LR in a sequence you, well first of all, you would backtrack, which would mean you would get a starting point, yes, and maby mathematically it’s sensical, yet doing something like that is not equivalent to rotating the sphere, it’s equivalent to being in a certain coordinate and going back to the start, the moment you seperate every movement in a sequence, is the moment it only becomes that, a sequence, not something physical.

    Just like you said, by cancelling out certain sequences you’re considering it as an independent piece, which logically, shouldn’t be considered that.

  7. Em

    There 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.)

    1. njwildberger: tangential thoughts Post author

      Em: 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.

      1. Em

        Do 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]
        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.

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  10. M. I.

    Let us consider the so-called “Law of Logical Honesty”:

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

    I think you violate this regularly. Can you make 2 independent choices? I doubt it. However, I don’t think you have ever defined what you mean by “independent” or “choice”. Set theory, which you frequently pooh pooh, does actually provide definitions of these notions. If you have defined these things, please reply here with references or definitions.

  11. Peter Tobias

    I find the question ambiguous. Do I find the logical steps of Banach and Tarski nonsense? Not in the least. They’re valid within their ZF+C world. Do I find their conclusion paradox? Most definitely. The ZF+C world seems to be quite far away from our own world. And I suspect that it was the intention of Banach and Tarski to show us just that.
    . . . What to do with their result? Most mathematician shrug, move on, and work within the ZF+C world, because proofs are easier there. I would prefer proofs within ZF+CC or ZF+DC (countable and dependent choice, respectively), because non-measurable sets don’t exist there and neither does the Banach-Tarski paradox. But I guess I have to look for those proofs myself.

      1. Peter Tobias

        First, ZF+C is clearly defined. Its axioms might have been born while Zermelo was on drugs, but the community of mathematicians has long poked at those axioms and found no contradictions within that model.
        . . . Second, mathematics is a mental construct, like chess or any other game. You can try to defame mental processes as hallucinatory or using any other pejorative adjective, but mathematics remains joyful for the users and useful for the world.
        . . . Third, you’re free to limit your own mathematical world to non-infinite entities or exclude the Axiom of Choice or exclude the zero from the Natural Numbers. If you do math right, the numerical results that you do get should be equivalent to the results of other mathematician. But to encourage other mathematician to adopt YOUR assumption, you have to do more than to defame their choices, you have to show that they at least risk getting non-equivalent results easily. I think Banach and Tarski did that to the Axiom of Choice.

  12. Sylvain

    > the community of mathematicians has long poked at those axioms and found no contradictions within that model.

    Isn’t the paradox precisely the demonstration of a contradiction in ZF-C? Even assuming that the intermediate sets have no definite measure, a sphere has mathematical properties whose measure is definite, such as its volume. Whatever the intermediate stages of reasoning, if we come to the conclusion that 1 = 2, isn’t this the fundamental criterion for rejecting a set of axioms?


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