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Since the Banach-Tarski paradox makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My reasoning is this:

If you can "duplicate" an abstract 3-dimensional ball defined, in the usual way, using the domain of real numbers, then clearly the domain of real numbers must be unsuited to describing physical objects (for instance The Earth), because duplicating them would double their mass (and thereby energy). But we use real numbers to reason about nature and derive new laws (or basic results) all the time in physics.

In the same way as we could integrate over the earth's volume to determine its mass from its density (for instance), we could apply the Banach-Tarski theorem to show that we could generate as many earths as we liked, right?

Does this mean that the mathematical foundation of physics is flawed? (I hope this question is not too provocative!)

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I've asked the question many times if real numbers might just be non-fundamental to the fabric of space-time, but I feel like I always get flamed for it. Oh Physics SE, people have argued to me that real-number geometry is a fundamental type of mathematics, as opposed to arising from observation. But I recognize that both relativity and quantum would have a problem with a universe operating on some Turing-complete program. I agree with that, so my answer is "no" the universe doesn't use real numbers but we don't know what it does use. –  AlanSE Feb 1 '12 at 17:21
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No, it doesn't. It just means we are curious creatures with fantastic imagination. –  Pratik Deoghare Feb 1 '12 at 17:32
    
The universe uses real numbers, but only uses sets which are measurable, is, I think, the moral to be drawn from the Banach--Tarski paradox. Anyway, functional analysis only uses measurable functions.... –  joseph f. johnson Apr 10 '12 at 23:54
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6 Answers

Philosophers and engineers have often thought that the notion of a real number is, indeed, unphysical, but this has nothing to do with the Banach-Tarski paradox.

One can find seven pieces of a sphere which, when re-assembled, produce a sphere twice as large. That is not strictly speaking a paradox, but it certainly violates one's geometrical intuition and must be un-physical.

The key point is that these pieces are not measurable: their boundaries are so complicated that one cannot meaningfully talk about their volume. It follows from this that the pieces cannot even be approximately produced by physically implementable cutting or slicing or powdering operations.... The notion of measure, as in probability theory, integration theory, and real analysis, is rather a mathematical technicality, so I won't go into it unless requested, but if you think of volume, you will get the right idea.

I will say that in more mathematically rigorous treatments of chaotic dynamics, such as Benatti, Narnhoffer, Thirring, also Connes, the physical dynamical variables, or observables, must be measurable functions on the phase space.

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Here's how I think about it.

Mathematics by itself, especially modern mathematics, says nothing about the world. It is a closed system, which we use so frequently in everyday life that we forget the distinction between the formal meaning and our physical interpretation.

For example, if you have a box with two apples and another one with three apples, together you have five. This is a true statement about the world, which results from our mental concepts of the terms "two", "three", "five" and "together". It is fundamentally different from the theorem $$2+3=5\ ,$$ which can be proven in the mathematical model of integers (ZFC framework). Because this model is so useful in describing how we naturally think about integers, we often mix between the two, but they are different.

In much the same manner, the Banach-Tarski theorem (it is not at all a paradox) is a statement about real numbers. It simply means that the mathematical model of real numbers has some weird consequences, which we generally do not interpret as "true". In the apples example, this would correspond to, say, finding that $2+3\ne 3+2$. If that has been the case, this would not have implied that addition is not commutative, but rather that the model is not perfect.

So, in short, the Banach-Tarski theorem says that how we think about volume is not well modelled in measure theory, but that was known from much simpler "paradoxes".

If you want to to be more blatant, you might say that it says that our basic intuition about measure is not consistent, because if assume that measure is translationaly and rotationally invariant, then there exist sets, that can be translated and rotated in a way that doubles the volume of a sphere. So measure was not preserved in the process, although each translation and rotation is measure preserving. This is formally possible because the individual sets are non-measurable, but violates our intuition about volume.

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The notion of the continuum in physics is not as a collection of differentiated points gathered together into an abstract set, but as a limit of discrete structures with finite computations defined on them. The limiting process must be well defined, so that the answer to any experimental question to any accuracy can be answered by a finite computation.

Any set-theoretic property of the collection of the real numbers which relies on separating out individual points from one another in a non-computable way and talking about them using logical properties of the individual points which involve undecidable questions is always going to be unphysical. This is true for much milder collections than those involved in Banach-Tarsky style constructions.

For example, consider the collection of all real numbers whose digits encode the solution of the Halting problem (the collection of all Turing degrees above 0'). There is a logical predicate which will describe these numbers--- there exists a computer program which takes the digits of this real number, takes a computer program, and spits out the answer to the question "Does the program halt?" after a finite number of steps. So this collection of numbers is well defined as a logical construction (at least in an ordinary set theory). This is a predicatively defined collection of real numbers, which makes sense as a set in mathematics.

But is it meaningful to formulate a theory where the electron behaves differently when it is located at one of these special points? Obviously not! The behavior of the electron in a physical theory must be described by a finite computation, not by an abstract logical game over all the digits of its position. The fundamental principle in physics is that computation is fundamental, not set theory. The idea that physics is computable is the foundation of the science--- physics looks for a computer program which matches the behavior of physical objects. It does not look for an axiomatic set theoretic structure which matches the behavior of these objects.

This principle has been sometimes challenged, but the challenges are mostly silly. It is not directly relevant to your question, because you are interested in Banach-Tarski.

The analogy of Banach Tarski with the Turing degree undecidability is actually awful, because the Banach Tarski paradox is of a far more non-constructive nature, in that it is not even possible to define reasonable properties of points which are in one or the other of these Banach Tarski sets. You can't find a predicate for separating out the points of these sets, even using super-duper-duper computation of any strength. Defining Banach Tarsky sets relies on a notion which is worse than computationally undecidable, they rely on a predicatively undefinable notion. You can't decide whether the points fall into this or that uncountable choice collection using any sort of predicate which does not refer to a choice function.

The only good analogy is not much of an analogy at all, the Banach Tarsky business is just identical in philosophical annoyance to every other axiom of choice construction where you apply the axiom of choice to a set of size continuum or higher. The countable axiom of choice is not a problem, nor is the axiom of choice on any uncountable sets you care to introduce which are not as big as the continuum. But the moment you can do uncountable choice on the continuum, you get paradoxes with probability.

The basic reason is that the notion of probability is intuitively well defined--- you can pick a real number at random by flipping coins at each step to pick the binary digits. But once the random real number is calculated to infinite precision, you can then ask "does it belong to this set S or not?" For every set S. This means that every subset of [0,1] has a notion of measure, which is the probability of a random number landing in that set.

This idea is the basic conflict between the axiom of continuum choice and the theory of probability. If you allow yourself to choose continuum many points, you can construct sets which are not measurable. If you do not allow yourself to do this, you can make every subset of [0,1] measurable. Whether you decide to have choice or probability is up to you, and most mathematicians choose choice over probability. This is stupid, and it makes measure theory difficult because you have to restrict yourself to measurable sets, and these measurable sets include every set you can ever imagine, including all the Turing degree shenanigans, and even higher undecidable stuff, excluding only impredicatively defined selections using continuum choice.

So I consider Banach Tarsky type results to be far worse than unphysical, they go so far as to be non-mathematical--- they should be considered false even as pure mathematics. This is to be distinguished from the concept of the real number whose digits encode the solution to the halting problem, which you can pretend exists in a real sense without any damage (although you still need to be careful to note its non-computable nature). There is no benefit whatsoever to including choice constructions on the real numbers, and there is much harm to the theory of integration and probability.

Computation in mathematics

Even within pure mathematics, the mechanism of logical deduction is always a finite computation. If you are given a well defined collection of axioms, or axiom schemas all of whose axioms can be listed by a computer program (this includes every reasonable mathematical theory), you can write a computer program to deduce all the consequences of these axioms. Godel's completeness theorem states that every deduction will be reached by the rules of first order logic, and that when there is an undecidable statement, one which cannot be proved or disproved by the axioms, there is always a model of the axioms where the statement is true, and a model where the statement is false.

This means that when you are given a set theory, which talks about infinite non-denumerable collections, you can understand that the theory is really talking about its countable models, and this gives a countable computational interpretation to every theorem. You can then ignore the jibber-jabber about the theory talking about some enormous sets, and consider the theory as talking about its countable models.

So, for example, when a theory is saying "all real number can be matched one-to-one to aleph-1", you can understand this to mean "all the countably many real numbers in any countable model of this theory can be matched 1-1 to the countably many elements of aleph-1 in this theory using a function symbol which is defined within the theory". You also know that "R is uncountable", meaning "for any function symbol in the model of the theory, mapping the integers to R, there exists a real number x which is not in the range of this function symbol." This does not make the real numbers in the model uncountable, of course, it is only saying that the theory is strong enough to prove the uncountability of R, so it can never identify the countability of the reals in any of its models from within.

Then mathematical theories never talk about non-denumerable infinities, except as a very useful figure of speech, and all questions about whether a theorem is provable or unprovable are equivalent to questions about countable structures whose properties are generated by an explicit computer program.

It is important to always have this point of view in the back of the mind, because it makes the undecidability results of set theories completely intuitive. If you think of the sets in set theory in Platonic terms, it is very difficult to make sense of the undecidability results, nor of large cardinals, or of anything else.

How Does Banach Tarski work?

In order to explain Banach Tarski, it is best to consider simpler constructions which predates it by half a century. The first such "theorem" is the well-orderability of R. This is done as follows:

consider the set S of all nonempty subsets of R. Choose an element from each member of S, that is, for every nonempty subset of R, pick one element. Now consider R. This is a nonempty subset of R, so you picked some element. Let this element be x(0). Now consider R-{x(0)} (the set R with x(0) omitted). You picked an element of this set, so call this x(1). Now consider R minus both x(0) and x(1). This is a nonempty subset of R, so you picked an element of this set, so call that x(2). Continue by induction to produce x(n) for all integer n.

Now continue the induction over ordinals. The first element you find past all the integers is $x(\omega)$, which is the element you chose from the set $R - \{x(n)|n\in Z\}$. Continue over every ordinal. It is easy to see that if you are ever get stuck in this inductive process, the reason must be that you have already listed every element of R, and this means that you have matched R to an ordinal.

The remainder of the proof is to show that there must be a large enough ordinal to make this process end. The reason is that if there weren't, then this process would bound all the ordinals from above, allowing you to bound the collection of all ordinals using a set, but you can't, because if there is a set of all ordinals, it is an ordinal, and you could define this ordinal plus 1 for a contradiction.

This proof, if looked inside a countable model, is producing a matching between the countably many elements of R, and some countable ordinal in the model. This matching is simply a fake--- it is revealing that both R and the uncountable ordinal it is matched to are countable in the countable model.

So now to Vitali--- to make a Vitali set, you consider an equivalence relation on the elements of [0,1) (considered as the unit circle, so that addition and multiplication are modulo 1), so that x and y are equivalent if their difference is rational (this is completely predicative). Then you magically choose one element from each equivalence class, and gather these into a set S. This set has the property that countably many translates cover [0,1), so this set cannot have measure.

This is a silly construction, because any attempt to specify which points belong to S requires an inductive listing of all these points. This is tantamount to an ordinal description of R. So you have a set which is basically a long ordinal list of real numbers, one for each equivalence class.

Banach Tarski does the same thing, except using translations and rotations. In order to make a finite number of sets cover the two spheres, it is essential to use non-commutative rotations. The argument is much more complicated, but the philosophical difficulty is the same as always--- the notion of choosing a real number at random is conflicting with the notion of choosing continuum many elements of sets simultaneously.

Forcing

The way to say this precisely was found by Paul Cohen. The method of forcing allows you to add elements to R in a countable model in such a way that they can match one-to-one with an ordinal which is bigger than any ordinal you like.

The basic idea is to choose a real number at random for each of the countably many elements of the ordinal you want to fit into R. This is not precise, because the notion of randomness is too complicated, so Cohen used a purely logical notion of choosing a "generic" real number, which is defined by the process which decides which properties are true of this number. The procedure is described from a nearly completely computational point of view in Cohen's "Set theory and the Continuum Hypothesis", and it is not described from a computational point of view hardly anywhere else.

From similar constructions (but this time using probability), Solovay proved that it is consistent to allow all subsets of reals to be measurable. The consistency of set theory plus dependent choice (strong countable choice) plus Lebesgue measurability of all subsets of R is one of the most striking results of set theory--- it guarantees that there is absolutely no Banach Tarski style paradoxes for sets constructed in the usual way, using predicative definitions.

In other words, if you do not allow functions which select continuum many elements at once, Banach-Tarski fails. There is absolutely no mathematical theorem which depends on uncountable choice which is of use to mathematicians, and it is well past the time to scrap this nonsense.

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Ron, Does not the Solovay model require assuming the existence of an Inaccessible Cardinal? If (mathematical) physics is to be described better with Solovay, then do you have an interpretation for this Inaccessible object? –  Roy Simpson Jan 28 '13 at 16:56
    
@RoySimpson: The "inaccessible cardinal" assumption is stupid: it's just the statement "there exists a model of ZFC", or equivalently "ZFC is consistent", and it is unprovable in ZFC because of Godel's theorem, but it's clearly a fine assumption, it's the standard "next step" beyond ZFC. You only need it to start with a full model of ZFC to show there are no contradictions once you add the random reals and then reduce to a submodel where all the sets you keep are measurable (the consistency of a measurable universe is enough to prove consistency of ZFC, so you need a small extra assumption) –  Ron Maimon Jan 29 '13 at 15:28
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This relates ZFC to ZFC-powerset by reflection, and ZFC-powerset is related to ZFC-powerset-infinity by reflection of the same type--- it's iterating a model-theoretic consistency statement as many times as ZFC-powerset-infinity has ordinals. ZFC-powerset-infinity is Peano Arithmetic (equiconsistently), which is a reflection of a different kind, over the integers, of primitive recursive arithmetic, and is completely well understood since the 1930s (with some more progress in the 1970s), it's described by the simple ordinal epsilon-naught. People were brainwashed that Godel killed Hilbert. –  Ron Maimon Jul 25 '13 at 5:49
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Thanks enormously, Ron. "The thing that is false in such a scheme is the absoluteness of uncountable ordinals" - this is an exciting statement and (although I'm not a real mathematician) seems to be a thoroughly original take on things. I certainly like the idea that finite computations are more fundamental than sets and by "constructivist" I probably emphasise less the intuitionist logic and more the idea of things being defined by a finite "computation" (constructive proof). –  WetSavannaAnimal aka Rod Vance Aug 9 '13 at 3:16
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Maybe the Babbage analytical engine gets in as the first Turing-complete computer defined, but you are absolutely right about the mechanism of the completeness theorem - I'd just never thought about it that way before, and, unlike Babbage its the first Turing complete computer whose creator specifically designed it with completeness in mind (I guess Babbage was striving for an undefined "generality"). I must look seriously into the Skolem theorem - another of my many projects! –  WetSavannaAnimal aka Rod Vance Aug 9 '13 at 3:46
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If you look close enough, you'll find that there does not exist a physical object fulfilling exactly the mathematical definition of a sphere. Therefore any mathematical statement about spheres only applies insofar to spherical physical objects as the approximation as mathematical sphere is valid. Indeed, even the concept of a solid body is just an approximation for macroscopic dimensions. Especially you cannot divide a body into two parts so that after division every point of the original body ends up in exactly one of the parts. Real-world objects always have fuzzy boundaries.

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Good question.

In my opinion, Banach-Tarski has absolutely no implications for the enterprise of describing physical reality i.e. physics. (It is still a good question though!)

Here's why.

Banach-Tarski just means that, okay, suppose I build a model of the actions that can be applied to a physical object. If that model has the property that every mathematically describable subset of an object can be acted on such that it gets translated independently of the rest of that object, then that model is very unlikely to describe physical reality. Well, that was quite obvious before Banach-Tarski; we don't need a measure-theoretic paradox to know that you cannot slice up a physical object in arbitrary ways and then translate those pieces completely independently of one another.

So, I don't think this paradox tells us much about physics at all.

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the Banach-Tarski paradox is impossible with any finite partition of the ball. If you think about that, it suggests that this paradox is an elaborate proposition equivalent to the fact that both the interval $[0,1]$ has the same measure, and hence is isomorphic to the interval $[0,2]$.

Does that mean a one-meter stick can be turned into a two-meter stick? No, because the equivalence is intrinsic to the continuum property. So any physical object will not be affected, because its made of discrete components.

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