This is a fabulous result, both positive and negative, for many reasons. But one of the things people don't realise is that there is a reason why it's interesting mathematically and not just a gimmick.
In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.
These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.
So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[0] cannot have all four obviously desirable characteristics.
For more information, here's a blog post[1] I wrote some time ago:
It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.
[0] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.
Nice write-up, though I wish you had expanded a bit more on this:
"We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product of an infinite collection of non-empty sets to be non-empty. That seems daft..."
Indeed, and it's exactly the sort of thing that you should not simply proclaim to be true with no explanation or reference.
>> "We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product of an infinite collection of non-empty sets to be non-empty. That seems daft..."
> Indeed, and it's exactly the sort of thing that you should not simply proclaim to be true with no explanation or reference.
I'll see if I can hunt out an accessible reference. Problem is, I think it uses Cohen's forcing, and that's pretty tough going in any kind of detail. I might add a note ... thanks for the suggestion.
You don't have to give a full proof, just an intuitive explanation of why the "obvious" arguments fail. For example, we can prove the non-emptiness of product sets for finite sets. Why can that not simply be extended in the "obvious" way to infinite sets? How can making the sets "bigger" suddenly make the product empty?
The problem I have is that "intuitive arguments" differ between people. What one person thinks is intuitively obviously true, another can think is intuitively obviously false.
I'll have to think about what I would be trying to say.
It's a good point ... it's now explicitly on my list.
>So we can use explicit examples to demonstrate that certain proofs are impossible
Nonsense. The independence of an axiom shows in general nothing about what you can and can not prove. A statement can just be false and therefore unprovable.
If you want an actual example of an impossible proof you want to look at the continuums hypothesis. With is probably unprovable and probably unable to be disproven.
>Banach-Tarski Theorem is a result that proves that a "Measure"[0] cannot have all four obviously desirable characteristics.
The important point is that it can't have these properties on all sets. If you restrict yourself to specific sets, these properties can be achieved. E.g. it is false that you can separate a sphere into countable Lebesgue measurable sets which can be rearranged to form a set with larger Lebesgue measure.
>> So we can use explicit examples to demonstrate that certain proofs are impossible
> Nonsense. The independence of an axiom shows in general nothing about what you can and can not prove. A statement can just be false and therefore unprovable.
You clearly know about these things, but I think you have not seen my intended meaning. It's easiest to use the example of Euclidean Geometry to clarify.
We have axioms A1, A2, A3, and A4. We want to prove A5 from these. But we can show that it is impossible to do so by creating a system (for example, spherical geometry) wherein A1, A2, A3, and A4 are all true, but A5 is false. The existence of this system then shows that A5 cannot be proven from A1, A2, A3, and A4.
Thus the explicit example of spherical geometry shows that it is impossible to prove A5 from A1, A2, A3, and A4.
I wonder if your deeper knowledge of the subject is leading you to expect more of my statements than what I'm actually saying.
>> Banach-Tarski Theorem is a result that proves that a "Measure"[0] cannot have all four obviously desirable characteristics.
> The important point is that it can't have these properties on all sets.
Yes, that is one of the four desirable properties.
* Measures 1 on the unit object;
* Countably additive;
* Isometry invariant;
* Defined on all sets.
We can't have all four of these, even in one dimension. But if we relax "Countable Additivity" to "Finite Additivity" then we can have all four properties in one dimension, and in two dimensions. What the Banach-Tarksi Theorem shows is that even with the relaxation in property two of Countable to Finite, in three dimensions we still can't have a measure with all four desirable properties. That's why we don't relax that property, we relax property four, and no longer require the measure to be defined on all possible (sub)sets.
> In general case given a system it is impossible to prove it is not self-contradictory.
If you're asserting it, by merely asserting axioms, then (in the general case,) yes. If you're constructing it, and showing that the axioms are satisfied, then no: there are several ways to construct a model that nobody has ever found a contradiction in, and you can construct spherical geometry this way.
> It really does. The existence of a system where A1-4 are true, but A5 is false proves that you can't conclude A5 from A1-4.
No it does not. First you'd need to show that your A1-4 == true, A5 == false system is not self-contradictory. Cause if it happens to be you basically proved A5 by contradiction (assuming A1-4).
It's impossible to prove that a (sufficiently-strong) system is consistent in that system. It's perfectly possible to prove that many systems are consistent with respect to the consistency of (say) ZF, and ZF's consistency isn't controversial.
All proofs are, ultimately, proofs to one's satisfaction, so bringing Gödel's incompleteness into this isn't insightful, imo. It's like bringing up the Problem of Induction in a discussion about science: yeah, you're right, but now we're not talking about the interesting thing any more.
> The axioms, which we have discussed in the previous chapter and have divided into five groups, are not contradictory to one another; that is to say, it is not possible to deduce from these axioms, by any logical process of reasoning, a proposition which is contradictory to any of them. To demonstrate this, it is sufficient to construct a geometry where all of the five groups are fulfilled.
Just formalise this section in ZF, and it drops right out.
1. Accept that our intuition about volumes is off when dealing with point clouds so weird that they cannot actually be described, but require the axiom of choice to concoct them.
2. Reject the axiom of choice and adopt the axiom of determinacy. This axiom restores our intuition about volumes to all subsets of Euclidean space, at the expense of which sets can be formed. (That said, the axiom of determinacy allows other sets to be formed which are not possible with the axiom of choice, so it wouldn't be correct to state that the axiom of determinacy causes the set-theoretic universe to shrink.)
3. Keep logic and set theory as it is, but employ locales instead of topological or metric spaces. Locales are an alternative formalization of the intuitive notion of spaces. For many purposes, there are little differences between locales and more traditional sorts of spaces. But, crucially, a locale can be nontrivial even if it does not contain any points. Locale-theoretically, the five pieces appearing in the Banach–Tarski paradox have a nontrivial overlap (even though no points are contained in the overlapping regions), hence you wouldn't expect the volumes to add up.
Why didn't you list the actual "standard" approach of accepting that certain sets have no definable volume? This is the basis of measure theory, which is perfectly accepting of the fact that measures don't need to be defined everywhere.
If we're doing dad jokes now, it's unfortunate that King Solomon didn't know about the Banach-Tarski paradox. Otherwise, instead of cutting the baby into two pieces, he could have suggested five pieces and made both mothers happy.
The gorgeous book "Discrete groups, expanding graphs and invariant measures" by A. Lubotzky investigates the structures that give rise to measure-theoretic phenomena like the B-T paradox. It is a graduate level monograph, but I recommend it wholeheartedly. It illustrates how the study of some paradoxes from the early 20th Century led to amazing and highly applicable mathematics like expander graphs and the spectral theory of non-commutative groups.
An aside, I have never seen anyone write the digit 8 like that before.
My takeaway from the video is that real numbers between 0-1 or 1-2 are infinite. And infinity+1 or -1 is still infinity. Even if you take something out from between 1-2, it will remain the same.
I was going to recomend that video. It has a good description of the technical details, but it uses a graphical representation to make it clear. But it's not a stupid graphical representation that hides all the details under flashy animations.
If you cut up the sphere's surface into pieces, the combined surface area will remain the same. If you then reassemble them in a different configuration into two spheres both the same size as the original, the surface area will be twice as much.
I don't see how that could be true. What am I missing here?
ETA: thanks for all the explanations. The most succinct answer seems to be because it assumes the surface is made of infinitely many points, and infinity breaks math. 2*inf = inf.
One more reason why it makes no sense to treat infinity as a number.
Take a (solid) sphere ... there are uncountably infinitely many points.
Now for each point colour it read, green, blue, yellow, or purple. Do this completely randomly.
Then every point will have nearby, arbitrarily close, other points of every colour. The colours will be deeply intertwungle, with the points of each colour just being a "cloud", spread throughout the entire volume.
This is a better mental image of what the "pieces" are like.
Now, if you do this colouring in a very, very special way, the red points can be rotated around to match exactly the green points. Well, OK, so the green points are a rotation of the red points. Not a problem.
But you can also arrange it so that if you rotate the red points a different way then they will exactly match the blue points. Well, OK, so the blue points are also a rotation of the red points.
The utterly, utterly bizarre result is that the above can be true, and we can also have a third rotation of the red points that will match all the green points and the blue points AT THE SAME TIME!
At its heart, that how the B-T theorem works. There are details, and arranging that this happens in also non-trivial, but at its heart, this is what's happening.
The purpose of the theorem is to show that the concept of "volume" cannot be applied to arbitrary sets of points in 3D. If you want more details about that, I wrote a blog post about this many years ago. It was my project for my BSc (Hons) way back in 1982.
The core of the paradox is that that the intuition that the volume of a bunch of disjoint sets obey the law
V[A ∪ B ∪ C ∪ ...] = V[A] + V[B] + V[C] + ...
is only guaranteed if you have a countable number of sets[1]. If you split a sphere into an uncountable number of pieces in the right way (which requires the Axiom of Choice) you can break this rule without being inconsistent with measure theory.
Completely wrong. From the Wikipedia article: "Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball."
There is absolutely no issue with uncountability here. The issue is with the particular shape of the parts, where V is not reasonably definable.
You first split the sphere into an uncountable number of subsets, then group these into a finite number of subsets, whose measure sum to twice the measure of the original set.
(Un)countability is at the core of most of the counterintuitive results of measure theory, exactly because of the the third property of measure.
To be clear, the construction given here violates the finite additivity property of measure. It's got nothing to do with the countable/uncountable additivity property.
IIRC correctly first you split the surface into an uncountable partition then you use the axiom of choice to "color" each point of each partition of a color and then define your finite number of subsets as all the points of each given color.
(Not an explanation of how it works, but just of how it could be true)
IIRC it follows a principle similar to the Hilbert hotel[0]. The idea is that you can split an infinity into an infinity[1] of infinite parts.
for a simplex example imagine all the points (n,m) where n,m are natural numbers, lets call this set P. we can split P into two sets:
- Q defined by all the (n,m) in P where n<m
- R defined by all the (n,m) in P where n>=m
Now you can "bend" Q by mapping (n,m) into (n, m-n-1) and R by mapping (n,m) into (n-m,m).
These bent version of Q and R are both identical to the initial P set.
This was a very informal and messy proof, but the core idea is the same: split the set (like a sphere surface) into many sets, manipulate (rotate) each one taking advantage of their infinity, recompose them as needed.
I do not think I am able to legibly comunicate the idea behind Banach-Tarski, but hopefully this gives some intuition
Essentially the difficulty arises from attempting to assign a measure (area) to every single subset of the sphere, where you say that rotations need to preserve this measure. The paradox can be viewed as a proof that you cannot assign a measure to every subset of the sphere in a consistent way.
The way measure theory resolves this is by showing that if you restrict to appropriate subsets, called measurable subsets, you can get all the nice properties you would expect.
It turns out that basically everything is measurable. In fact the existence of a non-measurable set is independent of ZF. This means that you need the axiom of choice, which was used here in the Banach-Tarski paradox, in order to construct a non-measurable set. So measure theory doesn't really lose a great deal by restricting in this way, which is why it gives such a great theory of integration.
The pieces are fractal and exceptionally complex. It’s related to the idea that a point has zero length, but a line (an infinite collection of points) has nonzero length.
It also highlights a discrepancy between our physical intuition of space and the way we model space mathematically.
It comes down to the fact that not every set can be reasonably assigned a volume. Once you restrict yourself to sets where volume is meaningfully defined, the paradox immediately disappears and your reasoning becomes completely valid.
It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.
So you chop up a sphere (which has a volume V1) into a finite number of collections of points (which don't have a volume), then assemble the collections of points into a new sphere (which has a volume V2 != V1).
The trick is to cut it up to an extent that you can't meaningfully assign area (or rather volume) any more.
The only real trick here is the low number of pieces, if you allow an arbitrary number of pieces it's simple to split the points into two sets and move the points into two equal sized spheres, that's just because infinity is weird that way (see Hilbert's hotel if you're not sure).
The question then becomes can you meaningfully talk about volume if you cut a sphere into a finite number of pieces? Turns out you can't.
Not really, Banach-Tarski is very similar to the hilbert hotel with odd-even rooms, the genius is in finding a way to recompose these parts using only rotations.
Not quite sure what part you disagree with, but indeed that's more or less what's happening. Though also the fact that it's a finite number of pieces is important.
> The only real trick here is the low number of pieces, if you allow an arbitrary number of pieces it's simple to split the points into two sets and move the points into two equal sized spheres
I took this as saying that it is easy to split the sphere into an infinte number of (I implicitly assumed) non null sets and recombine them in two spheres, to which I disagreed that it is hard to do so using only rotations.
But I guess what you meant is that you can build a bijective mapping S^2 -> {0,1} x S^2 just like you can build one for R -> {0,1} x R.
I did not consider the extreme case of every element of your partition being allowed to be a single point.
I'm with you here, doing finite -> infinite -> finite transformations logically creates paradoxes. If those infinite points have no dimensions, how do their cumulative zeros add up to a positive number?
To the risk of sounding naive, I see this as a self-inflicted paradox, much like the immovable object and the irresistible force paradox. If it's a paradox like Schrödinger's cat thought experiment, then I'm cool with it, because it points the limits of theory by carrying it into absurdeness.
While I haven't read the proof sufficiently in depth to explain it, but these aren't physical spheres because they have infinite parts.
They're less practically offensive to our senses, but I don't think this is really any different from zooming infinitely into a fractal or the difference between countable and uncountable sets. Infinities behave strangely.
I don't know if this will explain your questions, but years ago I watched this video from Vsauce about this very topic, and IIRC it was explained quite nicely: https://www.youtube.com/watch?v=s86-Z-CbaHA
The key thing is that this is not something even vaguely physically realizable. The pieces of the sphere are not "pieces" in any physical sense - they are just subsets of points inside the sphere, that don't have a definite shape and volume.
This must be the most unintuitive result of all of mathematics. Its very interesting what a seemingly simple axiom like the axiom of choice can lead to -- simple as in 'even a 9-year old can understand it', the consequences are rather enormous and not simple at all.
Honestly, it's not that surprising if you learned the properties of infinity before, especially of uncountable infinity. If 2*Inf == Inf, and if a sphere has an infinity of points, it's not that surprising that you can make two spheres from those same points. The construction itself is of course much more impressive, I'm not downplaying it, but I don't think it's less intuitive than other properties of infinity.
My personal reckoning with this was learning that there are as many numbers in the [0,1] interval of the real line as on the whole real line.
The BT paradox includes the requirement that the pieces are separated and put back together using isometries of R3, which is _way_ more restrictive than isomorphism of sets (what you're talking about). So it's quite surprising from that point of view!
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
Well, the axiom of choice gives a lot of counterintuitive examples, with the Banach-Tarski paradox being the easiest to imagine by a non-mathematician.
Yet, I know no consequences that would be measurable in physics. To my knowledge, AoC is more like glue, which (paradoxically) makes quite a few things smoother, e.g., all Hilbert spaces have a basis. Otherwise one runs in a lot of theorems, in all corners of maths, with "this is always true for finite, for infinite we know that there are no counterexamples, yet we cannot prove that for all cases".
> Yet, I know no consequences that would be measurable in physics
One can build a physical device modeled off of a Turing machine that enumerates all proofs within ZFC. The machine halts if an inconsistency is discovered, and runs forever if not. Now a prediction can be made about a process in the physical universe whose outcome depends on the axiom of choice.
I’m not trying to sound facetious actually. Highly abstract mathematics plays a critical role in inductive inference (in the sense of speeding up universal search by mapping a search over program space to a search over proofs in formal systems). This appears to be the direction some recent ML research is heading, so it wouldn’t surprise me if a lot of “unphysical” axioms end influencing our ability to efficiently approximate Solomonoff induction.
Perhaps ironically, despite appearances, the process you propose does not depend on the axiom of choice.
This is because we can prove, in the small and generally trusted metatheory PRA, that ZFC is inconsistent if and only if ZF (= ZFC − AC) is inconsistent (if and only if IZF (= ZFC − AC − LEM) is inconsistent).
[ This metaproof rests on the fact that ZF can prove that the axiom of choice (AC) holds in "Gödel's sandbox" L, the "constructible universe", even if it might not hold in the universe of all sets. ]
In other words: Adding the axiom of choice to ZF doesn't cause new inconsistencies. In case ZF is consistent (a statement which most logicians believe), then ZFC is so as well.
Indeed, and one can give specific metatheorems in this direction:
For instance, regarding statements of the form "for all natural numbers x, there is a natural number y such that %", where in "%" no further quantifiers appear, there is no difference between ZFC (Zermelo–Fraenkel set theory with the axiom of choice), ZF (set theory without the axiom of choice) and IZF (set theory without the axiom of choice and without the law of excluded middle).
Any ZFC-proof of such a statement can be mechanically transformed to an IZF-proof, with just a modest increase in proof length.
As far as we can tell, GR implies, and we have measured, space-time is completely continuous. Draw a square on a piece of paper; or, better yet, outline a cube with some sticks: within that square (or cube) is an infinite set of points of either the integral or real cardinality — whichever you’d like.
The “no physical infinity” thing sounds like a very Greek sort of axiom — like their “nature abhors a vacuum” thing, etc.
To my mind GR (or at least the standard textbook version of it, anyway) _assumes_ that space-time is continuous, it does not _imply_ it as such. Continuity is baked into the foundations of any physical theory that is expressed in the language of differential equations.
You probably know this, but it's easy to confuse the map (a physical theory) with the territory (reality, which is far more complicated).
No, physicists do not think that space is like a tinily subdivided grid. Lots of physics would break if that were the case, including QM.
We have a lot of hints that the amount of information in any given space might be bounded, but yet that space appears continuous. How exactly you reconcile this is (one of) the mystery of quantum gravity.
Sure, we cannot measure infinity, but to be fair, all mathematical concepts (when looked at closely enough) are not something we measure directly.
Even if a kindergarten-level maths of "there are three apples," we do an abstraction. We need to decide that something is a separate object, an apple (how big or small should a fruit be an apple? if there is a bite, is it an apple? etc, etc) - usually with an assumption that all apples are the same (which we know is not true, but serves as an useful approximation).
pretend that
> David Hilbert famously argued that infinity cannot exist in physical reality. The consequence of this statement — still under debate today — has far-reaching implications.
"Tame topology is the name for the largely programmatic quest for a refoundation of topology and geometry that avoids ‘pathological’ objects like space-filling curves or counter-intuitive results like the Banach-Tarski paradox that occur in the traditional approach."
In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.
These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.
So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[0] cannot have all four obviously desirable characteristics.
For more information, here's a blog post[1] I wrote some time ago:
https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheo...
It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.
[0] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.
[1] In case people want to discuss that separately I've submitted it as a separate post here: https://news.ycombinator.com/item?id=40798224