Once when I was talking about Banach-Traski theorem (paradox) I said:

OK! This is Banach-Tarski's theorem which is against our intuition but provable from our intuitive axioms! It says you can decompose a small diamond to finite parts and then attach them together and build a diamond as large as a mountain!

Suddenly one of the students happily said:

Amazing! Let's do it!

And I said:

Ah! Unfortunately you never can do it! Go, think and try on it! I offer a full mark to the best research.

At the end of the course Students' reasons were very confusing and strange! Some of their arguments were based on physics, some others related to philosophy, some inspired by religious beliefs, etc. I have my own reasons about impossibility of paradoxical decompositions in the physical world but I don't know if they are the best possible reasons or not. Here I want to ask about any possible argument which I can use to convince my curious students about impossibility of paradoxical decompositions in actual world.

Question. What do you say to students who want to apply Banach-Tarski theorem in practice?!

Remark. Note that if we use an argument which destroys the bridge between mathematical and physical worlds in the case of Banach-Tarski theorem, we should be able to answer a question in the following form:

We know many parts of mathematics which use Axiom of Choice in their main theorems (e.g. Analysis, Linear Algebra, Differential Equations, etc.) have amazingly correct applications in the actual world. Why Banach-Tarski theorem is an exception? Is it really an exception?!

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    $\begingroup$ I personally apply Banach-Tarski to increase the speed of doing dishes: I lecture my roommates on it during and suddenly their dishwashing capabilities double. $\endgroup$ – aureianimus Apr 7 '14 at 11:38
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    $\begingroup$ I'd say the first observation to make is that the dissection itself has no physical analog. Never mind the axiom of choice, ask them to start my taking an apple and separating it into two sets: the points with rational coefficients relative to some base and the points with one or more irrational coefficients. Then consider that use of the axiom of choice means that the B-T dissection is much fiddlier than that. $\endgroup$ – Steve Jessop May 2 '14 at 10:24
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    $\begingroup$ related: math.stackexchange.com/questions/103743/… We know many parts of mathematics which use Axiom of Choice in their main theorems [...] have amazingly correct applications in the actual world. Why Banach-Tarski theorem is an exception? Is it really an exception?! Much weaker versions of choice are sufficient to prove all the cases of these theorems that have real-world applications to physical processes. $\endgroup$ – Ben Crowell May 6 '14 at 21:15
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    $\begingroup$ Feynman's theorem says that the Banach–Tarski theorem has no practical applications: math.stackexchange.com/questions/250/…. :-) $\endgroup$ – LSpice Aug 29 '14 at 23:47
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    $\begingroup$ You already have lots of good answers, but let me point out what I think is the source of the problem, namely your statement of the theorem as "you can decompose...." As I've also pointed out elsewhere, the theorem is not really about anything you (or I or any other people) can do; it's exclusively about the existence of certain sets. $\endgroup$ – Andreas Blass Nov 8 '17 at 2:04

13 Answers 13


Whether one can or can not double the brilliant in real life has nothing to do with the Banach-Tarski paradox. Various mathematical objects are models of various aspects of our universe. $\mathbb R^3$ is typically taken as a model of three-dimensional space. But, it is only a model. Some things that are true in the model are false in space and some that are false in the model are true in space (just like any good model, it is not a precise replica of the thing being modeled, it distorts some things, neglects others, all so that it simplifies the original thing so that it is amenable to be studied by formal means).

The Banach-Tarski paradox is an illustration of (one of) the limitations of $\mathbb R^3$ as a model of the familiar (yet bizarre) ambient space we live in. There are plenty other such incompatibilities. For instance, perfect circles exist in $\mathbb R^3$, you may want to invite your student to construct one in real life. Physics seems to prohibit that just as much it prohibits doubling lumps of gold just by rearranging them piece-wise.

To conclude, models are models. If a model predicts something about whatever it models, then that prediction should be understood within the limitations and distortions made during the modeling process. An incompatibility between the model and reality may entail lots of things. Perhaps a need to abandon the model or a need to adjust it. But sometimes it should just be put away, stored on the shelf of those odd little things that the simplifications assumptions we made bring forth. To your students (or mine, as I often give this example in a first year calculus course), I would say: go ahead and decide how to categorize the Banach-Tarski paradox: is it a serious flaw in the model, a minor disagreement with reality, or an oddity.

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    $\begingroup$ +1. This exactly matches my thoughts on this question, and is a really good principle to keep in mind in general. $\endgroup$ – Nate Eldredge Apr 7 '14 at 19:11

This explanation from http://www.kuro5hin.org/story/2003/5/23/134430/275 might be useful:

One important difference between S and a real, physical sphere is that S is infinitely divisible. Mathematically speaking, S contains an infinite number of points. This is not true of a physical sphere, as there are a finite number of atoms in any given physical sphere; so a physical sphere is not infinitely divisible.

In fact, if we assume that spheres are not infinitely divisible, then the Banach-Tarski paradox doesn't apply, because each of the "pieces" in the paradox is so infinitely complex that they are not "measurable" (in human language, they do not have a well-defined volume; it is impossible to measure their volume). Immeasurable pieces can only exist if the sphere can be cut into infinitely-detailed pieces; this obviously isn't true for real spheres, since you cannot cut atoms into arbitrary shapes, especially not into infinitely complex shapes.

In other words; the mathematical "objects" are infinitely dense, but physical objects are not. Any physical analogies would involve division into two objects with the same volume but half the density of the original.

That is, Banach-Tarski doesn't really apply to physical objects, because the nature of physical objects puts them outside the domain of that theorem (Wrzlprmft's answer also describes this). You may have inadvertently opened the door to difficult questions from students by giving the brilliant => mountain analogy to begin with without prior explanation of its shortcomings. The closest result of that analogy would be a brilliant as large as a mountain but with the same mass as the original object, which isn't quite accurate to the theorem. To explain why it doesn't apply, you can touch on the differences between a mathematical "object" and a physical one wrt. infinite divisibility. To pre-emptively address the issues in the future, you can precede your physical analogy with a disclaimer that it is only a very rough analogy for the reasons given above.

  • $\begingroup$ Thanks for your answer. Welcome to MESE, Jason! $\endgroup$ – user230 Apr 7 '14 at 19:31
  • $\begingroup$ @SaintGeorg Thanks! $\endgroup$ – Jason C Apr 7 '14 at 19:38
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    $\begingroup$ and how do we know that real, physical spheres are not infinitely divisible? for all we know (molecules, atoms, particles, quarks), its turtles all the way down $\endgroup$ – rbp Sep 28 '14 at 21:54
  • $\begingroup$ @rbp The number of atoms (e.g.) in an object is countable. An object not being "infinitely divisible" does not preclude its atoms (e.g.) from being made of smaller components. $\endgroup$ – Jason C Sep 29 '14 at 5:57
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    $\begingroup$ @JasonC nor does it demand it, either $\endgroup$ – rbp Sep 29 '14 at 12:01

Matter is no bounded set of points with a non-empty interior

The sets in the Banach–Tarski theorem have to fulfill some requirements:

  1. They need to be bounded.
  2. They have to have a non-empty interior.
  3. They need to be, well, sets of points. In particular their elements are identical, i.e., do not contain any other information than position.

No matter how you interprete matter as a set (which is questionable on its own), a piece of real matter spectacularly fails to fulfill requirement 3, as it certainly isn’t homogeneous on the atomic level (which cannot be ignored for the inevitable non-measurable sets). Also, if matter is composed of anything particle-like, it is discrete set and thus requirement 2 is broken. Finally, quantum-mechanical probability densities do not have a bounded support and thus requirement 1 is broken.

  • $\begingroup$ +1 for clarity and actual address to physical properties. However, atoms are extraordinarily gigantic compared to mathematical points; why do you restrict yourself to the atomic level? A gold statue and a gold bar are certainly not interchangeable, yet they are homogenous at a lower level (atomic). Requirement 2 is more relevant as it seems very likely that fundamentally all matter does have only empty interior. Which brings up the question of spatial manipulation (of space, not of matter) which no one has mentioned. :) $\endgroup$ – Wildcard Jan 22 '17 at 16:00
  • $\begingroup$ Actually, I stand corrected: This MathOverflow answer addresses the relevance of space to BTP very nicely. $\endgroup$ – Wildcard Jan 22 '17 at 16:04
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    $\begingroup$ @Wildcard: A gold statue and a gold bar […] are homogenous at a lower level (atomic). – They are composed of the same types of atoms, yes. But those atoms are not arranged in a homogeneous manner, but form intricate crystal structures and have a complicated substructure. $\endgroup$ – Wrzlprmft Jan 22 '17 at 20:53

I think, the most important point is to emphasize that the proof relies on the axiom of choice. If this connection is made, you "only" have to convince the students that the use of the axiom of choice is not constructive.

I would ask them the following question:

Take an arbitrary subset $A\subset \mathbb{Q}$. Please give me a concrete constructive way to choose one element of $A$. Take into account that $A$ has in general no special element, i.e., no natural numbers, no minimal or maximal or "middle" element or similar. If you can give me a rule how to choose an element no matter how $A$ looks like, we can go on to $\mathbb{R}$, do the same or similar and then you could use Banach-Tarski.

The students should try things out, but all examples they give normally rely on the existence of a special element of $A$ and you can lead them to a contradition of their rule to find a choice function.

However, you should not be too direct. Otherwise the students will never want to use axiom of choice again.

Edit: As @mweiss pointed out in the comments, the question has an answer on $\mathbb{Q}$ (but not on $\mathbb{R}$).

  • $\begingroup$ (+1) Thanks for your answer. $\endgroup$ – user230 Apr 6 '14 at 15:06
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    $\begingroup$ Unfortunately I think your question (which is meant to illustrate the non-constructive nature of the axiom of choice) has an answer: Every elements of $A$ can be written uniquely in the form $m/n$ where $m, n$ are relatively prime integers with $n>0$. Choose the element of $A$ that meets the following conditions: (1) the denominator $n$ is minimal; (2) the numerator $m$ has minimal absolute value; and (3) choose the positive value of $m$ if there is more than one choice. This set of criteria specifies one and only one element of $A$. It's the "go on to $\mathbb{R}$" that's problematic. $\endgroup$ – mweiss Apr 6 '14 at 18:34
  • $\begingroup$ @mweiss Thanks! I will update that in my answer. But I think, from a pedagogical point of view, you can let them guess and maybe a student will get your solution and then you can ask: "Okay, $\mathbb{Q}$ is countable and here you have this special characterization of $\mathbb{Q}$; but how can you transform this idea to $\mathbb{R}$?". $\endgroup$ – Markus Klein Apr 6 '14 at 18:38
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    $\begingroup$ I don't believe the issue of being able to constructively choose (whatever that means) an element from any given nonempty subset of the reals is relevant. Even in the absence of the axiom of choice--any form, even countable choice--we can choose an element from a nonempty set as a step in a proof in ZF. $\endgroup$ – Dave L Renfro Apr 7 '14 at 18:53
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    $\begingroup$ @DaveLRenfro: I think your point is orthogonal to Markus Klein's. He's saying that choice isn't constructive. ZF says an element exists, but it doesn't say that you can define that element constructively. $\endgroup$ – Ben Crowell May 6 '14 at 21:28

Failing argument

Where one body is, there cannot be another. (Physical axiom)

If you wanted to decompose and recompose the brilliant, you'd have to move some parts of it through some others, which is physically not possible.

I don't know the proof, but I'm told, that there are decompositions and recompositions which avoid intersecting word lines.

Valid argument in classical physics

At least one of the decomposed sets is not Lebesgue-measurable. But the atoms of physical bodys are Lebesque-measurable.

Non-classical physics

I guess, it's not a paradox at temperature 0, but who knows.

  • $\begingroup$ (+1) About your physical argument based on Lebesgue mesurability of atoms, the point is that there are many known sub-atomic particles with and without mass. I am not sure how we should think about these entities but it seems we need a more complicated argument for refuting possibility of paradoxical decompositions in physics. Also I don't know if there is a relation between quantum uncertainties and this mathematical problem. On the other hand maybe a paradoxical decomposition is possible if we assume existence of more than 3 dimensions because we have more maneuver options. $\endgroup$ – user230 Apr 6 '14 at 14:52
  • $\begingroup$ @SaintGeorg As I said, maneuvering is not a problem. It's possible to maneuver the sets around in three dimensions. $\endgroup$ – Toscho Apr 6 '14 at 16:28
  • $\begingroup$ If you want to go non-classical, then it get's difficult. Uncertainties are surely no problem, as they just smoothen the Lebesgue-measure. Sub-atomic particles are indeed needed. The real problem lies in Thermodynamics, especially Bose-Einstein-statistics. $\endgroup$ – Toscho Apr 6 '14 at 16:31

It depends on the context of the course where Banach-Tarski is being presented. My answer relies on knowing a bit of measure theory. Let's call our intial set which we're going to split into some pieces $A$, and the reassembled version of the pieces $B$.

If $A$ is a bounded and measurable set in $\mathbb{R}^3$ and $A= \bigcup_{k=1}^n A_k$ is a pairwise disjoint partition of $A$ consisting of measurable sets, then the Lebesgue measure of $A$, $\mu(A)$ is equal to $\sum_{k=1}^n A_n$. This is the $\sigma$-additivity of the Lebesgue measure. Together with the translation invariance of $A$, this shows $\mu(A)=\mu(B)$, i.e.

the paradoxical example of recombining pieces of one sphere into two spheres cannot work if we restrict ourselves to measureable sets.

At this point, you can ask a (mean) counter-question: Construct a non-measurable set. This is quite hard; some courses in measure theory you show that there are actually non-measurable sets.

This way, Banach-Tarski shows how nasty and unintuitive non-measurable sets are. There's nothing wrong with the theorem, it's just that it shows how nice measurable sets are.

  • $\begingroup$ Thanks for your effort. Please see my comment below Toscho's answer. $\endgroup$ – user230 Apr 6 '14 at 14:56

1. One main point that has been made is that Banach-Tarski Paradox needs to use non-mesurable sets; I want to insist on this by comparing with another similar looking statements.

Theorem (Wallace-Bolyai-Gerwien). Given any two polygons of equal area, it is possible to cut one of them into finitely many pieces and rearrange the pieces into the second polygon.

Here the pieces can even be taken to be polygons. This theorem can be applied: you can produce examples in your Fablab. Explaining how different this is from Banach-Tarski is half the job of teaching Banach-Tarski, and I do not think that talking about the unknown possible future discovery in physics is more than barely relevant.

2. Another major point is that the pieces in the Banach-Tarski decomposition are not mathematically definable in any explicit way (we know that Banach-Tarski fails in some models of ZF: AC is really needed, not an artifact of the proof). Asking for the physical feasibility is at least one step beyond the true issue. And one should not let student think that the axiom of choice is just as any theorem; and in fact the Banach-Tarski paradox can be a good motivation to exclude it.

When you say:

We know many parts of mathematics which use Axiom of Choice in their main theorems (e.g. Analysis, Linear Algebra, Differential Equations, etc.) have amazingly correct applications in the actual world. Why Banach-Tarski theorem is an exception? Is it really an exception?!

you are missing an important point: the parts of analysis, linear algebra, differential equations that really use the axiom of choice have certainly no physical relevance or practical application either; I defy anyone to give one convincing one. These topics can all be derived without choice (but then not all linear space have a basis, of course).

  • $\begingroup$ (+1) Thanks for your answer. $\endgroup$ – user230 Apr 7 '14 at 19:54

We don't know if the axiom of choice is true or false. It might just be that in a black hole, you can actually appeal to the axiom of choice; on the other hand there are papers about counterexamples to quantum mechanics from certain failures of the axiom. And who knows, maybe in the future...

So the point that the axiom of choice is false in the physical universe is as valid as the point that the Banach-Tarski is impossible to do in the physical universe.

What is true is that we have no means to utilize the Axiom in the physical world. And since everything we are capable of doing has a volume, it doesn't allow us to make the infinitely precise cuts (over boundaries we cannot even define) that is required to make the Banach-Tarski theorem work.

At the end, my belief is that you shouldn't convince them it's impossible or false in the real world, but rather that mathematics talks about idealized balls in an idealized space; and not a pea or the sun in our physical space. Then ask them how they intend to construct something which is non-constructive, like non-measurable sets. Even if possible through some unknown physical phenomenon, it's unknown and we can't do it. For now.


When we talked about Banach-Tarsky in an introductory topology course back in the day, the professor explained it informally something like, "You can take a sphere and cut it into two pieces, and then reassemble those pieces to make two spheres of the same size!"

A student, looking a bit confused, asked, "How do you cut it?"

He thought for a moment and replied, "Very, very carefully!"

I thought this was both hilarious and a pretty good explanation at a layman level. (Others have explained the technical aspects of it very well above.) He included a few hand gestures that made it pretty clear to everyone that we weren't going to be able to make this cut with a pair of kindergarten safety scissors and a ping-pong ball.

At a layman level, that gets to a very basic reason why you can't do this in real life: You just won't be able to make the cut. For starters, the required cut is infinitely long. It's also infinitely curvy. Both of those are actually impossible in the real world.

During that same class period another student asked, "What do you mean when you say it's a paradox?"

"A paradox," replied our professor, "A paradox is . . . a surprise."

About as succinct an explanation as you're likely to find . . .


I'm baffled that no one has referenced the very entertaining book The Pea and the Sun, written for undergraduates and the possibly mythical "interested laypeople". Chapters 6 and 7 explicitly address this issue of physical (ir)relevance, leaving it to the reader in the end as to whether to imagine BTT in hadron behavior (not in making a pea into the sun, though).

By the way, the book also has more or less a proof in terms of words in groups (yes, it actually works written at this level), which would lead me to give the answer to a student that it's as much about that kind of abstraction as about nonmeasurable sets. Can you touch a rotation or flip in $D_4$? More likely you can only perform it. Same problem here.


I like this ...

from Spontaneous Phenomena, by F. Topsoe, Academic Press 1990:

THESIS 22: Those who seek a phenomenon which exactly follows a mathematical model, seek in vain.


Tell them that this seems like a cheap way of increasing the food supply by reproducing grain on a massive scale, but experience shows that the resulting loaves taste no better than sawdust so in practice world-wide hunger continues.


There are many who would say the same as the author of the following paper: "Pitowsky's attempts of hidden variables rather seem to reaffirm the old intuition: "if nontrivial set theory, non-constructive mathematics or a non-measurable set is used in an essential way, it cannot be physically relevant'." [Jakob Kellner: "Pitowsky's Kolmogorovian models and super-determinism", arXiv (2016)]

I would support this opinion, even if that makes set theorists angry. But in order to comfort them there are also applications like the following:

The Banach-Tarski Gyroscope is an intricate mechanism believed to have been constructed using the Axiom of Choice. On each complete rotation counterclockwise, the Banach-Tarski Gyroscope doubles in volume while maintaining its shape and density; on rotating clockwise, the volume is halved. When first discovered, fortunately in the midst of interstellar space, the Banach-Tarski Gyroscope was tragically mistaken for an ordinary desk ornament. Subsequently it required a significant portion of the available energy of the contemporary galactic civilization to reverse the rotation before nearby star systems were endangered; fortunately, the Banach-Tarski Gyroscope still obeys lightspeed limitations on rotation rates, and cannot grow rapidly once expanding past planetary size. After the subsequent investigation, the Banach-Tarski Gyroscope was spun clockwise and left spinning. [Eliezer Yudkowsky: "Banach-Tarski gyroscope", Hacker News (28 Dec 2008)]

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    $\begingroup$ The question is what you would say to students, now what opinion you support. Could you rewrite this to emphasize what you would say to students? $\endgroup$ – Rory Daulton Nov 12 '17 at 0:56
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    $\begingroup$ @ Rory Daulton: I have answered the question what I would say to students ( and what I tell them in fact every year) but the answer has been deleted with the advice that this site is not my soap box. Therefore I report here what others would say. $\endgroup$ – Wilhelm Nov 12 '17 at 13:26

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