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Banach-Tarski says that given a glass ball, we can break it into two glass balls of equal volume to the original (plus other generalizations). The explanations I have found for this paradoxical notion is that volume is not an invariant when we do these operations.

But, surely volume should be an invariant, right? "In reality," is it not realistic to expect that the volume doesn't change (unless we are dealing with some chemical property here, which I assume not since it is a mathematical paradox)?

So, my question is: is it possible, given whatever machinery we want to invent (something to break the glass exactly into what shape pieces we want, etc.), to actually perform this "paradox?" If not, then what is the "trick" behind the paradox which makes the math work out? What abstraction from reality do the hypotheses assume?

N.B. I've done extensive googling (especially on this site) of the subject and none of the answers have satisfied my question. I do realize there are a lot of questions about Banach-Tarski on this website already, but I do believe that my question is not a "duplicate."

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    $\begingroup$ The process of 'cutting up' is so pathological that you couldn't do it with a knife...? What can reality mean here. $\endgroup$
    – Frank
    Apr 14, 2015 at 17:25
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    $\begingroup$ In reality, the breaking will not be finer than atom or nucleus level or Planck length $\endgroup$
    – Hagen von Eitzen
    Apr 14, 2015 at 17:25
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    $\begingroup$ Where in reality you can find a continuous ball which is not made of atoms? When you find such a ball, and prove to me that it is in fact a continuous object, then we'll talk about coming up with the knife which can cut it into non-measurable parts. $\endgroup$
    – Asaf Karagila ♦
    Apr 14, 2015 at 17:59
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    $\begingroup$ @columbus8myhw: No, it means that as long as we believe that reality is made out of atoms, out of quarks and quantized particles, the question whether or not Banach-Tarski can be applied to solve world hunger by replicating oranges and other ball-shaped foods to everyone, is realistically meaningless. $\endgroup$
    – Asaf Karagila ♦
    Apr 14, 2015 at 18:26
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    $\begingroup$ I take the paradox to be one of many indications that our concept of the "coninuum" (i.e. the real number line) is a very artificial construct that doesn't correspond to physical, spatial reality. (I personally only "believe in" Turing-computable numbers.) $\endgroup$
    – Kyle Strand
    Apr 15, 2015 at 0:49

7 Answers 7

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The main crux of the Banach-Tarski construction is that you can break up a measurable set into non-measurable sets. Measure is, in a certain sense, analogous to volume. Non-measurable sets are somewhat strange and it can be shown that under mild axiomatic assumptions that it is consistent that there are no non-measurable sets. So in that sense, there is a "reality" where the Banach-Tarski paradox doesn't exist.

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  • $\begingroup$ I would also say that the non-amenability of $F_2$ is a significant part of the construction, though probably not the one people think of when talking about Banach-Tarski's violating physical intuition. $\endgroup$
    – anomaly
    Apr 14, 2015 at 17:26
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    $\begingroup$ And when all sets are measurable, you can cut a line into strictly more pieces than points. Yes, that reality makes sense! $\endgroup$
    – Asaf Karagila ♦
    Apr 14, 2015 at 17:57
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    $\begingroup$ @AsafKaragila, can you point me in the direction to learn about that last claim? I've been studying measure theory for a little while and that statement intrigued me. $\endgroup$
    – A. Thomas Yerger
    Apr 14, 2015 at 22:21
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    $\begingroup$ @Soke In math, all solids are nothing but an infinite set of points. $\endgroup$
    – MartianInvader
    Apr 14, 2015 at 23:58
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    $\begingroup$ Just to state clearly what I think is going on: the "mild axiomatic assumptions" mentioned do contradict the Axiom of Choice; is that right? $\endgroup$
    – Marc van Leeuwen
    Apr 15, 2015 at 8:39
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Indeed, the Banach–Tarski theorem is not "realistic" in the sense of applying to the physical world. However, I think the physical reason for this that you gave is not adequate, because there is no physical law of conservation of volume. Even without a chemical reaction, volume can change under pressure (especially for gases, but also a little bit for liquids and solids.)

The usual argument against the possibility of a physical realization of the Banach–Tarski theorem is based on the physical law of conservation of mass, not volume. However, I think that argument is also flawed. According to the principle of mass–energy equivalence, the process of cutting up a physical object and separating its pieces adds mass to the system if the pieces are attracted to one another (which is often the case with physical objects.) The reason is that the separation requires energy, which is equivalent to mass. See the article on binding energy for a more thorough discussion of this topic.

(Of course, the resemblance between the physical possibility of increasing the mass of a system of bound particles in this way and the mathematical possibility of duplicating a ball using the Banach–Tarski theorem is purely superficial.)

We shouldn't expect the Banach–Tarski theorem to apply to physical objects, simply because it makes no claim to apply to physical objects. It is about the notion of set, which is an abstract notion separate from the realm of physical objects. One can make limited analogies between sets and physical objects, and the Banach–Tarski theorem is one example among many examples of limitations to these analogies.

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  • $\begingroup$ Re the first paragraph: I'm not saying that it's impossible, but I'm ruling out the possibility of, say, the expansion of gases with pressure since we are dealing with a mathematical paradox here (and surely it wouldn't be a mathematical paradox if that's what the trick was!). And thanks, your answer was quite helpful - "One can make limited analogogies between sets and physical objects" indeed. $\endgroup$
    – MT_
    Apr 14, 2015 at 19:44
  • $\begingroup$ @Soke Ah, it seems like you are taking about connections between set theory on the one hand, and on the other hand, some idealized physical theory that includes a notion of volume (and says that every object has a volume that is preserved under physical transformations) but excludes some other aspects of the physical world like pressure. If you fix a mathematical formalization of this theory (which would have to be different from the usual mathematical formalization in $\mathsf{ZFC}$ of volume as Lebesgue measure, by Banach–Tarski) then one could ask mathematical questions about it. $\endgroup$
    – Trevor Wilson
    Apr 14, 2015 at 19:59
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    $\begingroup$ Honestly, I don't think the contribution of binding energy to mass is really relevant here. One could easily imagine performing set operations in a universe where there is no binding energy, if that would make a difference. I think your last paragraph gets to the point better. $\endgroup$
    – David Z
    Apr 15, 2015 at 4:58
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    $\begingroup$ @DavidZ I interpreted the question as being about connections between the Banach–Tarski theorem and physical reality. Binding energy is certainly relevant to physical reality, so a priori it could be relevant to such connections (except that, as I think we agree, there aren't any connections.) One could imagine a theory of some alternative physical universe without binding energy, and ask about its connections to the Banach–Tarski theorem, but OP didn't specify such a theory, so I didn't address it in my answer. $\endgroup$
    – Trevor Wilson
    Apr 15, 2015 at 17:14
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    $\begingroup$ @DavidZ I should clarify: I agree that "one could easily imagine performing set operations in a universe where there is no binding energy", but I think that one could imagine this in many different ways. Since the Banach–Tarski theorem is rather subtle, I think we should admit the possibility that its bearing (if any) on such an imaginary universe might depend on the details of this imaginary universe. I found it easier to stick to the actual universe in my answer. $\endgroup$
    – Trevor Wilson
    Apr 15, 2015 at 17:19
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The balls in question are definitely not made out of glass or any other material. Even if we were able to break them down into quarks and had the ability to reassemble them at will, we still won't be able to do this. It is misleading to think of the Banach-Tarski paradox in those terms.

The paradox addresses aspects of the usual formalisation of the continuum that don't fit very well with our physical intuition. In this sense, the Banach-Tarski paradox is a comment on the shortcomings of our mathematical formalism. However, once one stops thinking of the oh!-so-real numbers as describing reality in any sense, the paradox ceases being one and becomes an intriguing mathematical result.

When properly channeled, nonmeasurable sets can become a useful tool. For example the closely related notion of an ultrafilter is a powerful mathematical tool.

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I would like to add only a little to what has been said. User2566092 points out the unmeasurability of the sets we use. The fact that non measurable sets exists should give one pause for thought on its own I think.

The problems inherent in trying to cut a sphere like that in real life are multiple. For one the sets you are using are very much scattered. They are the representatives of orbits under a relation after all. Second to get the representatives we need to use the Axiom of Choice. That again shows something is not going quite right. If Axiom of Choice is necessary then we are not only choosing from too many sets already, those sets also pretty much have to be quite strange, for example you can't get a reasonable (definable without AC) well order on them.

Now before I give the impression that if we just got rid of AC all would be well in the mathematical world, let me point out that many (most) of the alternatives are often quite awful as well. For example if you want measurability of all sets you might be tempted to go with AD (axiom of determinacy) which seems nice you get all subsets of the reals are Lebesgue measurable, have the property of Baire, and the perfect set property. Unfortunately you also get consistency of ZF. Which is bad enough, but worse yet you actually get inner models some "very" large cardinals. Now this (to me) says that AD is in some sense much more insane then AC. You even get that some cardinals which are usually very large (measurable) are actually quite small $\omega_1$. You might want to take a look at https://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy which is were I cribbed some of these results from.

What I'm trying to say is, although AC gives us some pretty strange beasts. Its main rival doesn't do much better. If you just drop AC altogether you lose lots of things you really tend to want. Like the fact that sets have cardinalities and are comparable.

So really once you go past "small" finite sets you tend to get many strange results.

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    $\begingroup$ I admit that very large cardinals might seem "insane" to some, but why is the consistency of $\mathsf{ZF}$ an "unfortunate" consequence of $\mathsf{ZF} + \mathsf{AD}$? As a weird consequence of $\mathsf{ZF} + \mathsf{AD}$, I would mention the statement that there is an equivalence relation on $\mathbb{R}$ such that the number of equivalence classes is strictly greater than $|\mathbb{R}|$. $\endgroup$
    – Trevor Wilson
    Apr 14, 2015 at 18:00
  • $\begingroup$ @TrevorWilson Don't get me wrong I like very large cardinals just as much as the next person. Thinking about it now I suppose my real beef is with the ZF + AD implying consistency of ZF. To me at least it feels like you get "too much" from just a simple axiom which doesn't really even talk about the existence of any "big" sets. AD just feels as though it's super powerful and thus more suspect. I tend to believe in measurables but above that it always feels like someone might come up with another inconsistency proof. $\endgroup$
    – DRF
    Apr 14, 2015 at 18:09
  • $\begingroup$ @TrevorWilson But I agree with you that the very strange partition results (and thus cardinality issues through the prism of AC) are probably a better example of strangeness. $\endgroup$
    – DRF
    Apr 14, 2015 at 18:11
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    $\begingroup$ @TrevorWilson Good point.:) I just like WhoKnows?(ZF). $\endgroup$
    – DRF
    Apr 14, 2015 at 19:00
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    $\begingroup$ Let me add to the comment by @Trevor, and say that $\sf AD$ is the non-choice way of saying that $2^{\aleph_0}$ is sort of a large cardinal. $\endgroup$
    – Asaf Karagila ♦
    Apr 15, 2015 at 18:43
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It is not physically possible to demonstrate the Banach–Tarski paradox. The sets in question are very bizarre and can be best described as a distribution of points. A good example which is related, and is easily understandable is the Vitali set. There is a Wikipedia article for this.

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You're treating it as though volume is an undefined function from a subset of $2^\mathbb{R^3}$ to $\mathbb{R}$ that you assume satisfies the following properties:

  • All bounded subsets of $\mathbb{R^3}$ have a volume
  • Any subset of $\mathbb{R^3}$ that has a volume has a nonnegative volume
  • The volume of the unit cube is 1
  • Any isometry on a subset of $\mathbb{R^3}$ that has a volume doesn't change its volume
  • For any positive real number $r$, for any subset of $\mathbb{R^3}$ that has a volume, stretching it along the $x$ axis by a factor of $r$ multiplies its volume by $r$
  • For any two disjoint subsets of $\mathbb{R^3}$ that have a volume, the volume of their union is the sum of each of their volumes
  • For any sequence of disjoint subsets of $\mathbb{R^3}$ each of which has a volume, if their infinite sum exists, its volume is that sum. Otherwise its volume doesn't exist.

If you didn't define volume and prove it satisfies those properties, how can you prove that a way of defining it that satisfies those properties exists? Volume can be defined by triple integration but that doesn't define volume on all subsets of $\mathbb{R^3}$ because not all subsets of $\mathbb{R^3}$ are triply integrable. The reason energy is conserved in nature is because a splitting of a sphere into 5 pieces and reassembled into two copies of it can't ever occur in nature and that doesn't mean no such splitting exists.

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The paradox is about volume not mass. Is possible to produce even more than two glass balls that will equal the first that you broke because you can make a hollow sphere of glass therefore create an object of the same volume. It will have half the mass but will occupy the same amount of space.

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    $\begingroup$ More precisely, the paradox is about Lebesgue measure on $\mathbb{R}^3$ rather than volume or mass (which are physical concepts, not mathematical ones.) Although it is common to equate Lebesgue measure with volume or mass, the paradox illustrates one of the pitfalls of doing so. Also, the Banach–Tarski paradox is about balls (filled spheres) rather than spheres (hollow spheres). There is a version for (hollow) spheres as well, but mathematical spheres have no thickness and therefore no "mass" (measure); if we modify this version to thicken the sphere, then it will double the "mass". $\endgroup$
    – Trevor Wilson
    Apr 15, 2015 at 17:32
  • $\begingroup$ I should clarify that I'm talking about modifying the Banach–Tarski paradox to apply to spherical shells in the natural way. It is also true that a spherical shell can be cut up into finitely many pieces and rearranged into two shells with the same outer diameter and half the thickness, but this is not what the Banach–Tarski paradox refers to (and it is less paradoxical, in my opinion.) $\endgroup$
    – Trevor Wilson
    Apr 15, 2015 at 17:38
  • $\begingroup$ That's not actually how volume is defined. I know people define a given volume of water to be the same as the same volume of mercury because that's how it appears at the macroscopic level but it actually contains less matter. I can't unambiguously say it has a lower volume at the nuclear level because according to quantum mechanics, its particles don't have a defined location. $\endgroup$
    – Timothy
    Oct 20, 2016 at 23:39
  • $\begingroup$ It's a theorem of ZFC that a closed subset of $\mathbb{R^3}$ that's a sphere can be split into 5 pieces and reassembled into two copies of itself and your answer doesn't explain the flaw in thinking that it should not be possible. Water of a given mass is defined to have more volume than mercury of the same mass despite the fact that its nuclei are smaller but at the macroscopic level, a hollow sphere is defined to actually have a smaller volume than a solid sphere of the same outer diameter. Some people might also be incorrectly counting the volume of the air as part of the volume of sand. $\endgroup$
    – Timothy
    Jul 7, 2018 at 14:06

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