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.
>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.
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.
(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.
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 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".
"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."
[+] [-] ColinWright|1 year ago|reply
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
[+] [-] lisper|1 year ago|reply
"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.
BTW, you might find this interesting:
https://blog.rongarret.info/2023/01/an-intuitive-counterexam...
[+] [-] constantcrying|1 year ago|reply
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.
[+] [-] IngoBlechschmid|1 year ago|reply
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.
I tried to give a varied account on the axiom of choice at the Chaos Communication Congress once, the slides are here: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...
[+] [-] constantcrying|1 year ago|reply
[+] [-] ragtagtag|1 year ago|reply
Banach-Tarski Banach-Tarski!
[+] [-] throwaway81523|1 year ago|reply
[+] [-] ykonstant|1 year ago|reply
[+] [-] carlos-menezes|1 year ago|reply
[+] [-] smusamashah|1 year ago|reply
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.
[+] [-] gus_massa|1 year ago|reply
[+] [-] adastra22|1 year ago|reply
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.
[+] [-] ColinWright|1 year ago|reply
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.
[+] [-] emblaegh|1 year ago|reply
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.
[1]https://en.wikipedia.org/wiki/Measure_(mathematics)#Definiti...
[+] [-] afiori|1 year ago|reply
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
[0] https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...
[1] the idea is that if K and H are two infinities then size(H * K) = max(H, K) for reasonable definitions of multiplication and size https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_multi...
[+] [-] adroniser|1 year ago|reply
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.
[+] [-] soVeryTired|1 year ago|reply
It also highlights a discrepancy between our physical intuition of space and the way we model space mathematically.
[+] [-] constantcrying|1 year ago|reply
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.
[+] [-] bheadmaster|1 year ago|reply
[+] [-] pangratz|1 year ago|reply
https://www.youtube.com/watch?v=s86-Z-CbaHA&t=15m25s
[+] [-] shiandow|1 year ago|reply
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.
[+] [-] ASalazarMX|1 year ago|reply
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.
[+] [-] foota|1 year ago|reply
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.
[+] [-] djvdq|1 year ago|reply
[+] [-] tsimionescu|1 year ago|reply
[+] [-] vouaobrasil|1 year ago|reply
[+] [-] maze-le|1 year ago|reply
[+] [-] tsimionescu|1 year ago|reply
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.
[+] [-] stared|1 year ago|reply
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".
[+] [-] hackandthink|1 year ago|reply
https://ncatlab.org/nlab/show/tame+topology
[+] [-] aquafox|1 year ago|reply
[+] [-] unknown|1 year ago|reply
[deleted]
[+] [-] senorqa|1 year ago|reply
[+] [-] constantcrying|1 year ago|reply
[+] [-] RandomLensman|1 year ago|reply
[+] [-] unknown|1 year ago|reply
[deleted]
[+] [-] aaron695|1 year ago|reply
"A matter fabricator provides matter for thought" on the hub - DOI:10.2307/24987222 ( https://www.jstor.org/stable/24987222 ) [Early April 1989]
It made quite an impression as a kid. Even 30 years later I think about it every now and again.
[+] [-] unknown|1 year ago|reply
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