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May 19

Banach–Tarski paradox

It's immediately obvious that, as stated, the theorem can't be right. I took it to think overnight, and came up with, what I imagine, is a solid disproval. And because of how easily it is disproved, I came to think if it's some inside math joke, or are there people seriously thinking they are smart for exploiting imperfections to make ridiculous statements instead of fixing them.

Let's just walk over strong form of it, which reads as:

Given any two bounded subsets A and B of an Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of A and B into a finite number of disjoint subsets, A = A₁ ∪ ... ∪ A_k, B = B₁ ∪ ... ∪ B_k, such that for each i between 1 and k, the sets A_i and B_i are congruent.

First of all, it states it can use finite number of elements, that's nice but redundant, see below. Second of all, it states that matching elements are congruent, that puts major restrictions on the statement. And third of all, it says it works with any bounded sets, which is demonstrably false.

In order for two items to be congruent, they must by definition have same shape and same size, therefore same volume. In order for two sets to match, they must have same number of elements. Therefore, any attempts at using infinite sets would be contradictory to the nature of the theorem, that is that sets would precisely match. Additionaly, it means that individual elements will have defined and finite volume. Furthermore, same number of elements of same volume resolves to same total volume. Since subsets are constructed from original sets by partitioning, they must have same total volume as original sets, and it will not be relevant whether volume is well defined for individual elements or not - even if one were to use infinite set.

That establishes that this theorem can only work with finite subsets to begin with, and that both sets A and B have to have same volume.

Now I will further argue that the theorem can only work for pairs of sets of very specific shape. Unfortunately, I lack mathematical education to readily bring up proper terms, so instead I will make some local definitions.

Let "local curvature" be an inverse of a vector from center of a sphere which surface exactly follows local surface of a given set, to the point on that local surface, multiplied by normal vector in this point. So that it's larger in magnitude for curvier surfaces and smaller for flatter, is zero for a plane and is infinity with undefined direction for a point.

Let "net curvature" be a set of "local curvature"-"area" pairs for each distinct module of local curvature, integrated over entire surface area with that curvature, with zero pairs excluded. Objects bound by flat surfaces will have empty net curvature, and so are objects with a bump compensated by a notch of same exact curvature and area. Objects with curved surfaces that are not compensated for with surfaces of same area, opposite sign and equal magnitude will have non-empty net curvature.

Axiom: out of two matching surfaces, one is a fragment of another. Therefore, local curvature in every corresponding point of matching surfaces is identical. If local curvature wasn't identical, surfaces would have been divergent and therefore not matching.

Theorem: sectioning a set into two subsets by arbitrary surface doesn't change net curvature. Sectioning a set this way will produce two subsets that will have matching surfaces of same area where they were sectioned, by definition. Thus net curvature of a set consisting of those two subsets will not include that fragment of subsets surface area because they cancel out due to having same area and module but opposite signs, and the only remaining curvature will be that of the original set. It is therefore impossible to produce a subset with different net curvature than original set by sectioning.

If two original sets didn't had same net curvature, there inevitably will be fragments with surfaces that can not be matched to any remaining surfaces from another subset. Since partitioning doesn't changes net curvature due to producing two identical surfaces with opposite curvature and no net change, the residual area of specific curvatures that can't be matched to any surfaces in another set will remain constant regardless of specific partitioning operations performed. It is thus impossible to partition sets A and B with non-matching net curvature so that resulting subsets could be precisely matched. If you were to attempt sectioning vacant fragment to cancel out a parasitic surface, you would have just created another parasitic surface of same exact curvature.

Summarizing all of above, the theorem only works for objects with same volume and same local shape.

For example, consider sectioning square to match a triangle. They both would have empty net curvature. Or they both could have a bump of same size and curvature, which would resolve to same net curvature between the objects. You simply section off two sides of a square at an angle to produce a trapezoid, put two resulting straight triangles on the top, and section off the bump and put it in new location. Now consider a solid block with spherical void inside it, and a solid ball. No matter how you partition the ball, there will always be convex surfaces that you can't get rid of, that can not be matched anywhere on the block, and vice versa with concave surfaces of the block's void.

The question is, if above text resolves to mathematically correct statements, in which case it disproves most of the theorem. Raidho36 (talk) 00:43, 19 May 2016 (UTC)

Most of this text is not even wrong. Take, for example, the statement that if two sets are congruent, they must be finite. This is false. An example is two unit line segments in the plane. These are infinite sets (they contain infinitely many points; consider Zeno's paradox), but they are nevertheless congruent, they have finite length, etc. Next, the statement "they must by definition have same shape and same size, therefore same volume" presupposes that the sets have volume to begin with. The sets involved in the Banach-Tarski paradox are unimaginably complicated. In fact, it is quite literally impossible to construct them. Their existence is only guaranteed by the axiom of choice. The sets have a well-defined outer measure. And indeed these outer measures are equal to one another. But outer measure is not additive. If you have two disjoint sets, the outer measure of their union need not equal the sum of the outer measures. This fact, that outer measure does not satisfy properties that one normally associates with "volume" is rather surprising. Instead, there is a natural class of sets, the measurable sets, on which the outer measure behaves more or less the way one expects volume ought to. In fact, any set that can be explicitly constructed (without the use of the axiom of choice) is a measurable set. And indeed, the Banach-Tarski paradox is actually false if we confine attention to measurable sets only. So it is hardly surprising that any set you can manage to construct without any formal mathematical training cannot possibly satisfy the Banach-Tarski paradox. There are much more general sets, like fractal sets, that can also be constructed using lots of sophisticated techniques, but they too are measurable, and hence cannot be the mysterious sets referred to in the Banach-Tarski theorem. The decomposing sets of the Banach-Tarski theorem are non-measurable. These are very strange sets indeed, because all though we "know" they exist because of axioms in set theory, it is impossible to exhibit one explicitly. Sławomir Biały (talk) 01:00, 19 May 2016 (UTC)

First, apologies for not noticing this was the refdesk :-/

Second, I haven't read the whole thing, but the main point I think you're missing is that the pieces we're considering don't actually have a well-defined volume. I think Sławomir goes into more detail on that (I haven't read his contribution in detail either), but that's the first thing to realize. --Trovatore (talk) 01:21, 19 May 2016 (UTC)

Really, the best thing you could do, as a first step, would be to understand the Vitali set. It shows a very similar thing, by a very similar (but much simpler) technique. It breaks up the unit interval into a countable infinity of pieces that are just shifts of each other (wrapping around). That shows that the pieces can't have a well-defined "length".

Once you've grokked that, Banach–Tarski won't seem quite so strange anymore. The big difference is that the Vitali-set construction uses infinitely many pieces, whereas you can do Banach–Tarski with five. That is a difference, but ultimately not such a huge one. --Trovatore (talk) 01:58, 19 May 2016 (UTC)

I would first like to point out that I did not made a statement that congruent sets must be finite, I made a statement that they must have same shape and size, by definition of congruence. For some shapes or sizes volume is not well defined but it shall be equal nonetheless. If original sets A and B are equipotent with each pair being congruent, then they have equal total volume, regardless whether volume of individual elements is even defined. Since subset is generated from original set by partitioning, it must retain same total volume. The notion of total volume doesn't comes from sum of volumes of individual fragments, it comes from their congruence and equipotence of their sets, individual fragments may well be non-measureable. As per theorem, they don't have to add up to a complete shape, they just have to be congruent and their respective sets be equipotent. And finally, partitioning preserves area of surfaces that are not matching to any surfaces from another set and therefore belonging to non-congruent elements, such as surface of a ball that can not be matched anywhere on a cube, because partitioning of any set to create opposite matching surface simultaneously generates another such surface, and makes no net change in the area of surfaces that can not be matched.

I see where this coming from with axiom of choice, however it clearly violates some very basic principles that can not be sidestepped by clever definitions. — Preceding unsigned comment added by Raidho36 (talk • contribs) 10:44, 19 May 2016 (UTC)

If they don't have a volume, it doesn't really make sense to say that the volume is the same. It's neither the same nor different. There's just no such thing as the volume. --Trovatore (talk) 10:47, 19 May 2016 (UTC)

1) Subset is created by partitioning of original set, preserving the total volume. 2) Subsets have congruent elements, that is they are identical minus rigid transformations that make no change in shape or size. 3) Subsets are equipotential. To summarize, sets have same number of elements, which are all identical pair-wise, and therefore must have same total volume. Particular volumes of individual elements are not relevant.Raidho36 (talk) 10:56, 19 May 2016 (UTC)

I'm not sure exactly which subsets or elements you're talking about. The subsets I'm talking about are the ones you partition one ball into, and reassemble to form two spheres of the same size as the original. Are those the subsets you mean? Their "elements" are just points, and of course any two points are congruent. The "number" of elements is the cardinality of the continuum, and you can definitely have two sets that both have that number of points, but which do not have the same volume.

In this case, though, there is just no such thing as the volume of the subsets. So your argument cannot proceed. --Trovatore (talk) 11:01, 19 May 2016 (UTC)

Your argument appears to be self-defeating. First of all, you can't assemble anything from non-measureable sets, that would mean that either original object had no size, or it was infinitely large. If sets were measureable, then the whole counter-argumentation you put simply doesn't apply. Second of all, you're stating that they're all "just points", which by definition have zero volume, and the partition subset would therefore be infinite; the theorem clearly states it works with finite sets and individual elements that will therefore have finite size and defined shape. And finally you make contradictory statement, starting with that two sets of points with same cardinality can resolve to different volume, then immediately saying that sets of points (i.e. "this case") have no defined volume. Raidho36 (talk) 11:15, 19 May 2016 (UTC)

" you can't assemble anything from non-measureable sets". Sure you can. Take two non-measurable sets and form their union. There you go: a new set made out of two non-measurable ones. Here "non-measurable" has a precise technical meaning. It does not mean that we cannot estimate the volume of the set (using the outer measure), it means that the set does not behave in reasonable ways under the operations of assembly and decomposition. See Carathéodory-measurable set. Sławomir Biały (talk) 11:26, 19 May 2016 (UTC)

By "can't assemble" I obviously mean "physically", by assembly rather than making unions or whatnot. Raidho36 (talk) 11:30, 19 May 2016 (UTC)

I don't know what you mean, but it seems like you're relying on physical intuition rather than mathematical precision. That's no good. I've already explained in great detail why this theorem is not a "physical" theorem at all. It is about non-constructible sets. Also, please see Carathéodory-measurable set. You might want to read up on measure theory more generally before continuing in this vein. You can have collections of points with the same cardinality as a line segment that have any positive volume you like. You can have collections of points with the same cardinality as a line segment that have zero volume. You can have collections of points with the same cardinality as a line segment whose volume fails to be additive. This latter case is called a "non-measurable set" in this context. Sławomir Biały (talk) 11:40, 19 May 2016 (UTC)

I urge you to re-read my statements more closely. You will then see that I in fact am not confined by intuition, but rather use established mathematical entities as a backbone for my argumentation. Original argument was that you can partition a pea and assemble two copies. However, if non-measureable sets are used, it's impossible. Non-measureable sets can not be used to assemble a measureable set. Raidho36 (talk) 11:45, 19 May 2016 (UTC)

"Non-measureable sets can not be used to assemble a measureable set." False. See Vitali set. "Rather use established mathematical entities as a backbone for my argumentation." You talk about "curvature" and "surfaces". The sets in the Banach-Tarski paradox are not bounded by surfaces. Indeed, most sets have this property, like fractal sets. The tools of smooth analysis are not equipped to make meaningful statements about such kinds of subsets of $R^{3}$. Also, if "physically assemble" means something other than taking a union, you need to be specific. There is no mathematical definition of this term. Sławomir Biały (talk) 11:53, 19 May 2016 (UTC)

If you believe that some issue is being sidestepped by "clever definitions", please correct them. The issue here is that volume is a rather tricky thing to define. It seems like you believe that "volume" counts the number of elements. That is not true. For example, a very large ball and a very small ball have the same number of elements, because there is an explicit bijection between the two sets, but they have different volumes.

So the crux of the matter is this: How do you define the volume of a set? It really does matter how that term is defined here. You can argue that the definition is wrong, but then you need to supply your own definition. This is a lot harder than it might seem. For example, in the above, you are talking about sets as if they have smooth boundaries. What about sets like the set of points with rational coordinates, Cantor dust, or the Alexander horned sphere? How does your definition of volume handle those? Sławomir Biały (talk) 11:05, 19 May 2016 (UTC)

Wait a minute, what? The BTP doesn't say anything about volume, so how could it be "false for some definitions of volume"? The BTP talks about partitioning spherical balls and reassembling them after rigid motions, not about volume. --Trovatore (talk) 11:09, 19 May 2016 (UTC)

Yes, you're right. I've emended my post. Sławomir Biały (talk) 11:19, 19 May 2016 (UTC)

Yes I know bijection doesn't implies that elements are identical, however the theorem states so - it says there would be congruence between elements, meaning they have identical size and shape. I also know that volume may not be defined but it's not relevant because in the "same total volume" rationale, counting volumes of individual elements is completely avoided and only properties of entire set are considered, just like with "same curvature" rationale. This is why argumentation about undefined properties of individual elements doesn't apply. Raidho36 (talk) 11:23, 19 May 2016 (UTC)

"Congruence" does not mean "they have identical size and shape". It means that there is an element of the Euclidean group that maps the one set to another. It may or may not be a "theorem" that this implies the sets have the same "size" and "shape", but you would need to supply definitions of those words first. Sławomir Biały (talk) 11:32, 19 May 2016 (UTC)

Congruence defined as existence of an isometry between the sets. Since isometry preserves distances and is injective, congruence therefore requires that two objects would have identical geometry. Raidho36 (talk) 11:36, 19 May 2016 (UTC)

True, the restriction of the Euclidean metric to each set is preserved by the isometry. So what? You said above that this implies that the sets have the same volume, right? How do you define their volume? The volume is a number, that is not computed from the metric in a direct way. You need to define this number, if you are going to use it in a mathematical proof. Sławomir Biały (talk) 11:43, 19 May 2016 (UTC)

Notion of congruence as identity is important when considering set as a whole, however it's irrelevant when applied to individual elements. Consider a convergent sum function of a divergent function that generates an infinite set. Even if individual values within the set are not defined, its sum converges to a finite value. You can then argue that if another function produces equipotent set with precisely equal elements pair-wise, its sum must converge to same number. Raidho36 (talk) 11:53, 19 May 2016 (UTC)

This is a tissue of mathematical nonsense. Who is talking about "functions" here? We are talking about sets. It is true that a set can be defined by its characteristic function. For a non-measurable set, the characteristic function would also be not a measurable function. It would actually exist at every point (in fact, characteristic functions are equal to 0 or 1 at every point). But non-measurable functions satisfy basically none of the ideas of calculus that you probably want to apply. They are not integrable, not differentiable, etc. "Even if individual values within the set are not defined, its sum converges to a finite value." This is obvious nonsense. A sum of things that are undefined cannot be well-defined. How do you define the sum without having the terms of the sum? Further, I have no idea what the sum of individual values of a set has to do with congruence, volume, or the Banach-Tarski paradox. And I suggest that you probably don't either. "You can then argue that if another function produces equipotent set with precisely equal elements pair-wise, its sum must converge to same number." Again, no idea what you're trying to say here. But equipotence (that is, having the same number of elements) does not imply equal volume, something you have asserted without proof on several occasions in this discussion. I will not be correcting you any more. If you have no wish to learn mathematics, then there really is no purpose to this posting. Sławomir Biały (talk) 12:14, 19 May 2016 (UTC)

I should add, before you get carried away with that challenge, that there already is a very general theorem characterizing the Lebesgue measure (and thus the outer measure). The Lebesgue measure is the unique way of measuring certain "obvious" sets (the Borel sets), which is countably additive on those sets, invariant under the Euclidean group, assigns a value of 1 cubic unit to the unit cube, and transforms under the group of invertible linear transformations by the determinant. So, roughly, any "reasonable" definition of volume on the Borel sets is the Lebesgue measure. Sławomir Biały (talk) 11:19, 19 May 2016 (UTC)

Is it fair to say that this theorem is true in the mathematical realm but not true (or not applicable) in the real world? Or is it somehow true that with proper, very advanced technology it could be implemented in practice? if indeed it's only mathematically true than people would probably be less offended by it if they understood this..??68.48.241.158 (talk) 11:35, 19 May 2016 (UTC)

It has nothing whatsoever to do with the physical world; the concept of an unmeasurable set is completely aphysical. (I do wonder when "argue with a crank" became an accepted reference desk past time, though.) --JBL (talk) 13:26, 19 May 2016 (UTC)

okay, I suspected this but I'm not a mathematician..I've read about this topic on a couple of occasions in the popular press..it might be useful to briefly explain this in the article's intro section for people who arrive there via pop reference..I might put it in the talk page there..68.48.241.158 (talk) 13:56, 19 May 2016 (UTC)

I just had a look at the current intro of Banach-Tarski paradox, and in my opinion the third and fourth paragraphs do a good job of describing the situation. Not that it would be impossible to do better, necessarily, but I don't think anyone could read the intro and come away with the conclusion that they can go home and cut up and orange into two oranges. --JBL (talk) 20:56, 19 May 2016 (UTC)

I think the intro could be improved, but the problem is finding the right RS discussions to cite. Here is a nice bit from this [1] thread on real-life application of B-T: "The Banach-Tarski paradox is an illustration of (one of) the limitations of ℝ3 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 ℝ3, you may want to invite your student to construct one in real life." If you can find any RS discussing the non-physicality of this paradox I can help you incorporate that into the article. SemanticMantis (talk) 21:06, 19 May 2016 (UTC)

I do wonder why you think insulting askers and implicitly criticizing other math desk responders is acceptable - it is not, and it does a disservice to our reference desk. SemanticMantis (talk) 16:00, 19 May 2016 (UTC)

Agree. And anyway the connection between maths and physics is not at all an obvious one, in fact is rather a mystery. Not quite up to mysteries like what is thought but getting there. Dmcq (talk) 16:08, 19 May 2016 (UTC)

Yes, there's not enough evidence to conclude that the OP is a crank. It seems more likely to me that he (I'm just gonna go with "he") is in over his head and unwilling to admit it. See Dunning–Kruger effect. He might be in danger of turning into a crank, though. I'm willing to bet this is how a lot of cranks get their start.

As for "arguing with" — it's my feeling that when you start to sense someone's in over their head, it doesn't do a lot of good to throw outer measure at them; you have to engage them at the level they have the background for (if you engage them at all) while of course not telling lies to children. --Trovatore (talk) 19:34, 19 May 2016 (UTC)

The whole thing was literally sketched on a napkin during lunch break, so it didn't accounted for simple fact that the theorem is not confined to measureable sets. The two described limitations still apply if you want to produce measureable sets by partitioning, and are easy way to test if one (semi)physical object can be re-arranged into another. Raidho36 (talk) 21:51, 19 May 2016 (UTC)

Sławomir Biały mentioned this earlier, but I think it's an important example to show how this kind of reasoning doesn't apply. Consider a solid three dimensional ball A with volume V. Now let B be the set of points in A with three rational coordinates and let C be all the remaining points. Obviously the union of B and C is the original ball A and has volume V. What is the volume of B and C? Both B and C are dense and might superficially appear to have volume V, but in that case volume isn't additive. You can't apply intuition to things like this. CodeTalker (talk) 20:16, 19 May 2016 (UTC)

There are mathematicians who reject the idea of non-measurable sets and who aren't crackpots. See, e.g., Constructivism (mathematics). -- BenRG (talk) 22:24, 19 May 2016 (UTC)

Constructivism means a lot of different things to a lot of different people, so it's hard to generalize, but just at a very initial level, constructivism doesn't play that nice with partitions of sets in the first place. In Errett Bishop's model, for example, there are no discontinuous functions at all, whereas the characteristic function of one of the partitioned subsets must be discontinuous.

There are constructive set theories, or at least ones that some people who call themselves constructivists accept. I don't know an awful lot about them, but my impression is that they tend to be rather weak, and not easily adaptable to this sort of question.

I think a lot of people look at this sort of question and they say, oh, uses the axiom of choice and uses sets that aren't explicitly constructed. Who doesn't like that? Constructivists. But the picture is a lot more complicated than that. --Trovatore (talk) 23:16, 19 May 2016 (UTC)

There is a Solovay model that satisfies ZFC and under which all sets are measurable and axiom of choice fails. I originally thought the theorem may have flaw in its construction, however I came to think that it can't be that simple and the flaw roots deeper than that. After all, paradoxes don't truly exist and if you encountered one it only means that the models you were using are flawed and need amending. If Solivay model is correct, and only measurable sets exist and/or axiom of choice fails, then the theorem immediately breaks down. And it should, because as it stands, it yields absurd results that are obviously wrong. This theorem is meant to be disproved, through improvement of theoretical basis and elimination of open loopholes that allow this kind of behavior to exist at all.

Now that I established that taking jabs at the theorem is pointless, I shall ask a few related questions to shed some light on the whole thing.

1) Is it correct that when you deconstruct set A into non-measurable subsets (which don't have defined Lebesgue metric) to produce derivative set A' that consists of resulting subsets, then set A' wouldn't have defined Lebesgue metric either? If so, then by what mechanism when you use set A' to reconstruct set B, its Lebesgue metric becomes defined?

2) The theorem postulates that [even] non-measurable subsets from sets A and B are congruent pair-wise. Congruence defined as existence of isometry function between two sets. Isometry defined as distance-preserving injective function. Is it correct that if set doesn't have defined Lebesgue metric (particularly geometric size), then distances can't be preserved and therefore mapping function would be simply injective, but not isometric? If so, since there aren't isometric mapping function between paired subests, they are not congruent. More generally, non-measurable sets are not congruent to any other sets, if previous statement is correct. Raidho36 (talk) 23:21, 19 May 2016 (UTC)

I'm not even going to get to your questions; I'm going to address two things in your first paragraph.

• It's a "paradox" in the sense that it violates your intuition. Sometimes, this means the theory is wrong and needs amending. Other times, it means your intuition is wrong and needs amending. This is an example of the second case. There's nothing wrong with using intuition. It's your intuition, meaning you personally, that needs to be changed.
• As for the Solovay model being "correct" — I think you may not be all that familiar with the use of the word "model" as used here. This is not a model in the sense of "a proposed way of accounting for things", as used in physics. It's a model in the sense of model theory. What it shows, in this case, is that ZF by itself, without choice, cannot prove the BTP. But no one (certainly not Bob Solovay!) thinks that the Solovay model is the true universe of sets. --Trovatore (talk) 23:32, 19 May 2016 (UTC)

If you use properties-preserving function on a set then resulting set must have preserved properties of the original set, by definition of the function you were using. If it doesn't, then it's a paradoxical situation of type I as per your description. You should address the questions though, they must be simple enough to answer. Raidho36 (talk) 23:50, 19 May 2016 (UTC)

They are simple enough, I think, the reservation being that I'm not entirely sure I follow the questions. You are not using language in a standard way and I may guess wrong as to what you mean. To be honest, I'm not sure you have the background to follow the answers. You need to learn some real analysis before it's even worth going into it in this kind of detail. It's not something you can learn in fifteen minutes or on the back of a napkin.

Still, I'll take a crack at it: If you partition A into sets A₀, A₁, A₂, A₃, A₄, which are not Lebesgue measurable, and then you look at copies A'₀ etc under rigid motions, then the A'_n copies are also not Lebesgue measurable. Is that what you meant by question 1? It's hard to tell, but it's my best guess.

Second part of question 1: Just because the pieces are non-measurable doesn't mean that their union is non-measurable. I don't know why you think it should mean that, so I don't know how to help you fix your intuition here. But it just doesn't mean that.

Question 2: There is no reason you can't have an isometry between two sets that aren't Lebesgue measurable. This gets back to the "background" question — I don't think you know what Lebesgue measure is, and as I say, you can't understand it in fifteen minutes. It is not the same as a metric; you seem to be mixing up the two things.

Not mixing up, just using wrong term. So the answers are "no" and "no", that's also simple enough. I must assume the Lebesgue measure of union of non-measurable sets can change because reasons. I get that it is not defined for individual subsets so they are not constrained by anything of the sort, and a union function can therefore produce any set. As a rhetorical question, what defines what set union operation will generate? The reason I think non-measurable sets couldn't have isometry comes from my background, where it is a convention that if something is undefined, it's not equal to anything, not even itself. By that definition of identity between undefined entities, it's impossible to construct isometry between non-measurable subsets since there's no injection function between them, they simply can never be matched. That's of course in addition ot previous statement that if geometry is not defined, the mapping function can not preserve it since it doesn't exist to begin with. Raidho36 (talk) 00:25, 20 May 2016 (UTC)

"[I]f something is undefined, it's not equal to anything, not even itself." That's exactly right! That's why you can't make any argument that goes through the volume of the pieces, because they don't have a volume.

Why you think this has anything to do with the existence of an isometry is still obscure. My best guess is still that you are mixing up the notions of measure and metric. An isometry is a bijection that preserves the metric. It must therefore also preserve measure, for sets that have a measure, but there is no obstacle to having an isometry between sets that don't have a measure. --Trovatore (talk) 02:20, 20 May 2016 (UTC)

I have abandoned the idea of preserving measure when I realized the theorem works with non-measurable sets. It is still important that metric is preserved, so the set must have defined metric spaces to begin with.

Anyway, I had too much time to think lately and I eventually came across a thought experiment: if you take an infinite ribbon, and fold it on itsefl infinitely many times but such that it is not allowed to intersect with itself, do you get a solid block? If you fold it uncountably many times? Ribbon is constructed from an infinite plane by sectioning. To put it another way, what is the "thickness" of a set constructed from a folded surface that's not allowed to intersect with itself? More generally, is it correct that if a surface is not allowed to intersect with itself, then folding operation must put the folded part on points immediately adjacent to ones that the surface already occupies? Raidho36 (talk) 22:03, 20 May 2016 (UTC)

The maps can still be measure-preserving; they'll preserve the measure of any measurable set to which they're applied. The A_n don't happen to be measurable, but the map might be (say) from all of R³ to R³. It carries A_n along for the ride. Any measurable subset of R³ will have an image under the map with the same measure as the original. Even if you restrict the map to A_n, there are still measurable subsets of A_n (maybe measure 0, but measurable), and their images will have the same measure as the original.

As for your "ribbon" question, we did that very recently. See Wikipedia:Reference desk/Archives/Mathematics/2016 May 7#how can lines/curves of infinite length have positive area?. --Trovatore (talk) 22:24, 20 May 2016 (UTC)

Thanks, but I will require elaboration. Does this mean that a flat surface has defined finite thickness, even if infinitely small? Raidho36 (talk) 22:32, 20 May 2016 (UTC)

Depends. Is zero finite? I mean, obviously zero is finite, but somehow physicists seem to have trouble figuring that out :-)

The "width" dimension of your ribbon is superfluous; it's easier to think of it as just curve in the plane. And the answer is that, even though the curve itself has zero "thickness", nevertheless it can have positive area. These curves are, of course, very odd, and fail to satisfy some of the attributes you would expect the English word "curve" to carry with it. Similarly, the "pieces" in Banach–Tarski fail to be entirely piece-like in a natural-language sense. However, unlike the B–T "pieces", these "curves" can be defined completely explicitly, and the argument does not require the axiom of choice. -Trovatore (talk) 23:56, 20 May 2016 (UTC)

If a curve have positive area, then by adding a dimenstion, we get that a plane has positive volume. The thought experiment was constructed to establish whether individual points have positive volume. It was important because I derived such definition of measure that implies that individual points have finite sigma-additive volume. Can you please deconstruct it and highlight where what is wrong with it?

Let fractional measure of set A be a known measure of its superset S consisting of all points bounded by it, multiplied by a number that describes what fraction of S set A constitutes.

Any subset is already a fraction of its respective superset. I reason that there must be a number that describes what fraction of points from a superset is in the subset. And indeed, if you bisect a solid unit sphere, you'll get two hemispheres, each of which contains exactly 1/2 of points of original shpere, and each have exactly 1/2 the volume of it. If you can construct a superset with known measure such that you also know what fraction of points are used to build the set you're "measuring", you can derive it's equivalent volume. In an earlier example, consider a solid unit ball that you partition into two balls such that ball A has points with rational coordinates and ball B has the rest. Intuitively, the balls must have smaller volume than original ball because they're both missing some of its points. Using fractional measure, you can establish that they must in fact have smaller volume precisely because they both don't have 100% of original ball's points. Since, as I imagine, between any two infinitely close rational numbers there is infinitely many real numbers, rational ball's fraction must be infinitely small and real ball's fraction must be infinitely close to 1, so rational ball's volume is infinitely close to 0 and real ball's is infinitely close to volume of original ball. Since you can partition a solid ball such that it's every subset contains exactly one point, and they must add up to the volume of original ball since they are all a certain fraction of it, then individual points must have defined, finite and sigma-additive volume. Which sounded absurd to me since that would mean that surface of a set has its own volume and volume of a closure of the same set would be greater by that amount. But I reasoned that if a point didn't had any volume, you must have been able to produce infinitely many disjoint sets containing this point, as per "stacking" thought experiment, which is a contradictory statement and thus the clause automatically resolves to "false". Additionally, since fractional measure is sigma-additive, then there must also exist fractional measure for sets assumed to exist in Banach-Tarski theorem, such that they will add up to the volume of the original set, so you wouldn't be able to partition sets of different volumes such as that they would have produced subsets with identical total fractional volume. This seems logical so I hope you can point out flaws in all of this, most notably resolve the whole "points have defined volume" thing, because I've been assured they do not.Raidho36 (talk) 22:32, 20 May 2016 (UTC)

A ball is not a countable set. Its measure is not the sum of the measures of its points. Although the measure of any countable set is zero. An idealized ribbon of zero thickness is a rectifiable surface, so must have zero volume. You cannot construct a non-rectifiable surface by folding a ribbon. Sławomir Biały (talk) 00:29, 21 May 2016 (UTC)

This is exactly why thought experiment was constructed. If a ribbon can't intersect, its fold can't lie in exactly the same points, it must therefore lie on adjacent points, and the resulting object, instead of remaining flat, is now 2 points thick. It will become twice as thick every time it is folded. Original ribbon may have zero volume, but resulting object would be a solid block consisting of all points in its region. That implies that each individual point must have defined positive measure. If they didn't, the object would have remained flat, which would have violated the definitions by which it was constructed.

Also, consider that this definition of measure makes it sigma-additive for all sets. I would really like to know why it isn't valid. Raidho36 (talk) 00:47, 21 May 2016 (UTC)

Isn't this a case of trying to figure out what 0 times infinity is?Naraht (talk) 00:52, 21 May 2016 (UTC)

There is no such thing as "adjacent" real numbers. And there is no continuous one-to-one function from a ribbon to a cube. Sławomir Biały (talk) 01:11, 21 May 2016 (UTC)

It's more like trying to figure out if what is defined as 0 is actually 0. It makes no practical sense since there are no two "adjacent" real numbers indeed, if there's always infinitely many between. Anyway, this doesn't answer the question, what happens if you stack infinitely many surfaces such that they may not intersect, but there also may not be free space between them? Do they simply lie infinitely close, while leaving infinitely many points in between, in violation of definition of function that constructed it? As a different view, if you partition a solid unit cube into parallel flat surfaces, will you find such points that belong to none of the surfaces? From definition of partition, you will most definitely not. That would mean that surfaces aren't just infinitely close together, there must be such pair of surfaces between which there are no points, on some not intuitively concievable level, "past infinity" so to say. Raidho36 (talk) 10:59, 21 May 2016 (UTC)

Sure, you can "stack" infinitely many surfaces and fill out a cube. The cube is already a stack of squares, one for each real number in the unit interval. But you wanted to do this with a single connected surface, and I'm saying that this is not possible without creating self-intersections. Also, there is no such thing as "infinitely close" in the reals. Sławomir Biały (talk) 12:42, 21 May 2016 (UTC)

The ribbon seemed like a good base at a time, disjoint surfaces really is a better approach. But anyway, does it all mean that each individual real has positive additive measure? Even if individual surface "has no volume", they still add up to a volume of a cube. I'm coming to realize the problem has a lot to do with the way measure is defined, and it must be defined in a way such that it exists and is positive for every non-empty set in its respective topological space, so that this kind of loopholes are eliminated. Tangentially, does that also means there are in fact "adjacent" reals if you can create a set such that every real belongs to unique disjoint subset and there are no reals between them? This is a very obvious contradiction, that on one hand between any two reals there's infinitely many other reals, but on the other hand there exists a set such that there's no reals between its subsets. Raidho36 (talk) 13:33, 21 May 2016 (UTC)

No, individual points have zero measure. Countable collections of points have zero measure. The real numbers are not countable. Seriously, the best way to get these questions answered would be to read a book on real analysis. Royden, "Real analysis" starts at the absolute basics. Sławomir
Biały 14:21, 21 May 2016 (UTC)

I already know that all of the points mentioned, but I came to belive all of it is simply due to measure being improperly defined. Lebesgue measure can only be not greater than some "true" measure, but is not guaranteed to be equal to it. It is not sufficient to just sweep such glaring imperfection under the rug and pretend it's not an issue, there must be a search for the "true" measure and Lebesgue measure as defined must not be accepted as a last instance measure of a set. So it's not a valid statement that "points' actual measure is zero as per Lebesgue measure" because it's simply not guaranteed to be actual. Raidho36 (talk) 22:45, 21 May 2016 (UTC)

The Lebesgue measure is certainly not the only measure available. For example, the counting measure assigns a value of one to singleton points. But it does not assign finite measure to things like balls and spheres, so you proposed argument does not lead to a contradiction. There are other measires, but as I said, the Lebesgue measure is uniquely characterized by certain reasonable properties. If you want to consider other measures, you need to forgo one or more of those properties. One can characterize all measures that are invariant under the Euclidean group. So, it's not a question of looking for something that hasn't been discovered yet. If you feel that one of these other measures is the "correct" measure for your argument to work, the you need to specify what it is. It is not the "volume", because that is given by the Lebesgue measure. Period. You aren't going to be able to find a measure that assigns the same measure as the Lebesgue measure to balls and is countably additive. It's a theorem. You can find its proof in every textbook on measure theory. Sławomir
Biały 23:31, 21 May 2016 (UTC)

Well that's my argument, that the set of data upon which it all hinges contains a flaw somewhere and needs to be amended such that it's eliminated. Just because quantum physics and general relativity don't work well together doesn't mean that the entities upon which they operate properly obey different laws of physics, it only means that they're both imperfect and need amending. It doesn't necessarily implies that there is a discoveries left within them that can lead to such amendments that it would then work on both quantum and macroscopic level, they may be both wrong (although approximately correct within some limits) and a whole another theory is right. There may be flaws in axioms because they're assumed on some intiutive level, which isn't necessarily correct. E.g. it's not correct to intuitively assume that out of two clocks, if one is running slower, the other is running faster. Raidho36 (talk) 01:40, 22 May 2016 (UTC)

The axioms of the real numbers cannot be "correct" or "incorrect". If one has a different set of axioms, then one is not discussing the real numbers any longer. Certainly, that is allowed. But the Banach-Tarski theorem is about the real Euclidean space, and real Euclidean group. The BT theorem is a true theorem about this situation. If there is another set of axioms that more closely models your intuition, nothing stops you from writing them down and proving that some analog of the BT theorem is false. But of course that would not invalidate the BT theorem, or any of the many other things that are true about the real numbers that are not true in the Raidho36 numbers. Sławomir Biały (talk) 02:56, 22 May 2016 (UTC)

It's important that Raidho36 understand that this is a theorem of ZFC, and it follows inevitably from ZFC's axioms, and intuition doesn't matter.

But don't tell them that their intuition is wrong and needs to be changed. Their intuition is fine, unless they want to work in set theory. One might even call it correct. -- BenRG (talk) 06:38, 20 May 2016 (UTC)

No, that intuition is indeed wrong and needs to be changed. This is a real fact about geometry, not just set theory. It can't be realized physically, as far as we know, but that is a different matter. --Trovatore (talk) 06:48, 20 May 2016 (UTC)

I don't know where that belief comes from, but it isn't from anything we've ever learned about the real world through the evidence of our senses. -- BenRG (talk) 07:02, 20 May 2016 (UTC)

It's the "realer" world than the physical one. See mathematical realism. --Trovatore (talk) 07:26, 20 May 2016 (UTC)

Oh god, here we go again :-D. But seriously, although non-measurable sets are not "constructible", is there any way to have even a picture of one in mind, to guide one's intuition? A fractal, for example, is described by some kind of similarity on different scales. Is there any analogous way to picture the essence of a non-measurable set, in human terms? This was why I thought about Hecke algebras, that there is some essentially arithmetic aspect to groups like R/Q. The book on Borel equivalence relations that you referred me to doesn't seem likely to lead to any useful geometrical intuition. Sławomir Biały (talk) 11:26, 20 May 2016 (UTC)

There's definitely some interesting mathematics involved in trying to understand non-measurable sets. For example, one of the standard constructions of a non-measurable set is to consider the abelian group $\mathbb {R} /\mathbb {Q}$. This is a bit of a weird group to try to understand: it consists of equivalence classes of real numbers x and y, where we identify two reals if they differ by a rational. So this space has an infinitesimal character: every open interval containing 0 has a complete set of representatives of this equivalence relation. In terms of radix expansions, we identify two real numbers if their difference has a terminating base n expansion in some integer base n. A choice of a single representative of each equivalence class defines a subset of the real line, which is non-measurable. In terms of groups, this is a splitting of the epimorphism $\mathbb {R} \to \mathbb {R} /\mathbb {Q}$.

A basic trend in mathematics is that instead of trying to study a space directly, it is often better to look at functions on the space. Measurable functions, for example. Because Q acts ergodically, any measurable function on R invariant under Q is constant. I believe this implies that the algebra of measurable random operators on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathbb R/\mathbb Q} is a type II von Neumann algebra. I believe there is some Hecke algebra construction as well. Is there an account of this anywhere? Sławomir Biały (talk) 23:48, 19 May 2016 (UTC)

This is a very good starting point for the study of Borel equivalence relations. Excellent references are Vladimir Kanovei; Borel equivalence relations. Structure and classification. University Lecture Series, 44. American Mathematical Society, Providence, RI, 2008. x+240 pp. ISBN 978-0-8218-4453-3 and Su Gao; Invariant Descriptive Set Theory. Chapman & Hall/CRC Pure and Applied Mathematics. ISBN-13: 978-1584887935. --Trovatore (talk) 09:11, 20 May 2016 (UTC)

Thanks for that. I'm afraid that treatment is a bit orthogonal to my own interests, but it looks like the paper "Ergodic equivalence relations, cohomology, and von Neumann algebras" by Moore and Feldman, referenced there, looks readable to me. Sławomir Biały (talk) 10:17, 20 May 2016 (UTC)

Actually, this example is discussed in part I of Murray and von Neumann from the point of view of von Neumann algebras. I haven't been able to find any treatment from the Hecke algebra perspective though. Sławomir Biały (talk) 10:42, 20 May 2016 (UTC)

Reality

One comment that I've heard on the Banach-Tarski paradox is that Reality is simply a crude approximation to some set of Mathematical Axioms. To me, the paradox is simply a close cousin to the fact that the even integers and the odd integers both have the same cardinality as the integers (which is similar in that the set is cut into two pieces which can both be mapped back to the original). What I think throws people off is that the "volume" of the sphere is bounded, *but* the mapping in the paradox uses the fact that there are an infinite number of zero-measure points, and comes up with a cool way to both cut them into 4 groups (actually a few more, in the exact proof) such that there is always a point in the old sphere that can map to anyone you need in each of the new spheres. Since in all cases, a point that maps into sphere A has a point that maps into sphere B less than epsilon away, looking at the volume of the subsets before they get mapped really doesn't lead to anything useful.Naraht (talk) 00:29, 21 May 2016 (UTC)