Banach-Tarski and the Paradox of Infinite Cloning
quantamagazine.org
quantamagazine.org
Yes, this stuff is fishy, and yes we can blame ZFC which is a bad formalization in comparison to what we've developed since. But the real scandal is why does our definition of geometry "leak" the underlying set theory it's built atop so much? Surely it's bad to have such a leaky abstraction in pure math!
The series goes on to show that by abandoning "points" — which pull all the funny set theory stuff into geometry/topology/whatever is the topic at hand, one can still have a classical foundation — e.g. with the axiom of choice and law of excluded middle — that makes mathematicians feel at ease, but also purge this Banach–Tarski gobbledygook.
Yeah... Those crazy HoTT people, trying to actualize the goal of putting mathematics on an actually firm foundation and removing the rest of the gobblygook handwaved into the religion of math as opposed to the pure logic it represents...
Also you can use HoTT WITH AC / law of excluded middle... It's just not there by default and there are some really nice things you get without it, so it's pretty much only the lazy crutch of mathematics since forever. If you see proof via excluded middle, consider it a code smell (and recall by the Curry-Howard correspondence the proof is essentially code)
[0]: https://en.wikipedia.org/wiki/Homotopy_type_theory#Special_Y...
[1]: http://math.andrej.com/2016/10/10/five-stages-of-accepting-c...
I think things like the Banach-Tarski theorem are the other side of that coin: they're showing some of the places where the formalisation we're starting with isn't a great fit for some things we might hope to use it for.
I don't think I'd go as far as to say that makes the formalisation outright bad, but looking at alternate systems which don't admit Banach-Tarski-like results is surely a worthwhile way of spending time.
See https://golem.ph.utexas.edu/category/2021/06/large_sets_1.ht... for tackling the "large cardinal pissing contest" that is much of modern set theory.
Your very statement is a good retreat from platonism with blinders, acknowledging the inherit "moral relativism" that there are many possible foundations, and it is up to usflawed humans to decide what we like to work with best.
The earlier intuitionists like Brouwer were polemicists, perhaps because they felt very alone. Now there is a good network of CS-mathematician hybrids to keep everyone feeling more sane.
Here we see the dual track that you can question your foundational choices and your higher level abstractions (point-set topology vs locales which are distilled to being purely order-theoretic) concurrently. It's nice to take the same skepticism and interest in finding definitions the work with not alienste the working mathematician at multiple levels.
Because, for all the trepidation about abandoning ZFC, the mainstream formalizations have clearly failed in that mathematicians that aren't logicians or set theorists would rather engage with them as little as possible.
PhD mathematician in industry here. The way I see it, foundations is to the rest of mathematics the way music theory is to music: it needs to be a describer, not a prescriber. (If I were less charitable I'd have said "ornithology is to birds").
> the mainstream formalizations have clearly failed in that mathematicians that aren't logicians or set theorists would rather engage with them as little as possible
On the contrary, ZFC has been a tremendous success in that most mathematicians don't need to worry about it at all.
One can say that if either is inconsistent then they can prove everything, but that makes it even sharper: the large cardinal is used purely to give engineering assurance of software correctness and not real mathematical rigor. So it's a pure engineering use of one of the most "out there" mathematical objects. It doesn't seem worse than using IEEE floating point arithmetic to design airplanes....
That sounds nice, but breaks down when one thinks harder. Music is a little bit physical phenomena, a little more biological phenomena, and even more cultural phenomina. That's many layers at once, and theory has to conform to the evidence.
Math, is not science. This is no evidence external to reasoning. Different foundations / formal systems conclude different things.
At best, we can look at what matches existing working mathematicians mental heuristics and.....that's not ZFC, which admits all sorts of crap because it is untyped.
> On the contrary, ZFC has been a tremendous success in that most mathematicians don't need to worry about it at all.
That is how most mathematicians see it, but us in the type theory crowd see that as bad goalposts necessitated by the fact that ZFC is so clunky to work with --- of course one wants to declare mission accomplished and move on to other things as quickly as possible with a foundation like that.
Check out https://xenaproject.wordpress.com/ for a less heterodox approach, that nevertheless does use a type theoretical "user interface" and "kernel" (trusted foundation) for purely practical reasons. Basically, the idea is making making formalized mathematics not a huge burden necessitates a more ergonomic system than was needed 80 years ago without computers.
> I think things like the Banach-Tarski theorem are the other side of that coin: they're showing some of the places where the formalisation we're starting with isn't a great fit for some things we might hope to use it for.
I don't follow. You can view the Intermediate Value Theorem as something that motivates the definition of "continuous function", so that once you have the definition it had better conform to the theorem, sure.
But the Banach-Tarski theorem isn't like that. It's just a cool result of some other things that work well. It's not motivating anything or being motivated by anything.
So it isn't parallel to the intermediate value theorem, but opposite to it.
Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski”
Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want.
You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axioms, we get nonsensical results.
In general, it seems like we’ve picked _pretty good_ axioms that mostly give us sensible and useful results. But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere.
The only way you're going to avoid getting results like this is with axioms like "there is no such thing as an infinite number". At that point, the real line doesn't exist (too many points) and it becomes impossible to duplicate spheres by dividing them at a level of fineness that also doesn't exist.
But that's not a productive approach to anything.
But the Twitter link at the top of this thread seems to have a rather more interesting way of doing so.
The Banach-Tarski theorem is not the only theorem out there that bothers some people. Anything to do with infinities gets a large number of outraged rejections.
OK, which axiom do you want to replace?
Actually selecting and proposing an axiom set is way outside my knowledge-base. My limited understanding is that the Axiom of Choice, in ZFC leads to Banach-Tarski, and if it’s removed Tarski doesn’t hold, but I don’t have nearly enough information to say if that’s worth exploring removing it.
The Cartesian product of non-empty sets is itself non-empty.
The Cartesian product of sets S_1, S_2, S_3, ... is of course the set of tuples (s_1, s_2, s_3, ...) such that s_1 ∈ S_1, s_2 ∈ S_2, s_3 ∈ S_3, ... . An element of the Cartesian product is a tuple with one element drawn from each of the sets being, um, Cartesianly multiplied.
Thus, the Cartesian product of the three sets {1, 4}, {a, b}, and {@, 2} is the set {(1,a,@), (1,a,2), (1,b,@), (1,b,2), (4,a,@), (4,a,2), (4,b,@), (4,b,2)}.
The Cartesian product of the three sets {1, 4}, {}, and {@, 2} is {}, the empty set, because no tuples exist such that the second element of the tuple belongs to the set {} (the second Cartesian factor).
So all the Axiom of Choice asserts is that, if all of the Cartesian factors are nonempty, then a tuple exists with one element drawn from each of the Cartesian factors. The only way for it to be impossible for such a tuple to exist is if one of the factors itself has no elements.
It's a theorem for finite Cartesian products, so all the dispute is over infinite products.
It's probably also worth mentioning that the C in ZFC stands for the Axiom of Choice, which is an indicator that people have explored not using it. ZFC without the Axiom of Choice is ZF.
Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite.
Happy to hear disagreements tho.
What this means is that a universe that contains infinities is, even in theory, entirely indistinguishable (in finite time) from an universe that contains really large/small but finite quantities.
That depends on the model of computation you pick, doesn't it?
Not holding my breath for either.
Even whole numbers are an abstraction that makes sense only when you can clearly define what is the thing you're counting.
The complex numbers (well, at least those with a rational imaginary part and a rational real part) have been recently proven to be necessary to describe the universe[0] (assuming quantum theory is correct).
The irrational numbers are then are the only numbers that are harder to pin down, and I'm not sure that there is a way to prove that any physical quantity has an irrational value, vs a rational value that is arbitrarily close to that irrational value.
I'm talking about using the abstract concepts of infinity as a useful mathematical tool to produce predictions. Notable example: calculus
You only need the exact number Pi if you want to measure something like the ratio between the length of a perfect circle and its radius with infinite precision. But you can't be sure your measurement has infinite precision with a finite number of measurements, and so you can't observe the difference between a perfect circle and a many, many sided X-agon, even if perfect circles do exist in the geometry of the universe.
Just as a fun aside, even if perfectly circular shapes do exist, it's unlikely that perfect circles would exist in physical objects - at best, you would have ellipses, and there is no (known?) way to compute the ratio between the length of an ellipse and the properties of its foci.
You're talking about measuring something in the real world. Measuring is basically counting how many thing a given reference thing fits into the thing you're measuring. I have no problems with the assumption that for all intents and purposes we live in a finite (space and time) physical universe and there is a maximum precision that will ever be necessary.
What I am talking about is that in order for the math we use to describe that universe to work out we need irrational numbers; otherwise you couldn't be able to prove theorems and whatnot. I think this makes the irrational numbers (e and pi in particular) quite fundamental tools and I don't care if the real world doesn't allow objects (or positions) to be measured with irrational numbers.
> there is no (known?) way to compute the ratio between the length of an ellipse and the properties of its foci.
There is no closed-form expression for the circumference of an ellipse. There is an infinite series though. Same for a circle; there is no closed-form for computing pi either!
Also wouldn’t your argument also apply to zero? You can never know if a quantity is zero as opposed to some enormously small epsilon that you haven’t detected yet. Is zero “unscientific?”
The reals however are a different problem, and it's not scientifically possible to prove that the ratio between the length and radius of any object is exactly pi (that it is a perfect circle). However, it's also impossible to prove scientifically that it is 3 or 3.14 or any other number.
Now my use of "unscientific" is more of a hyperbole or click-bait. I thought I explained my actual claim pretty well - that you can't measurably/scientifically distinguish between a universe that contains actual infinities and one that only contains some arbitrarily large numbers.
There's a difference between something not being instantiated in this universe and being unscientific, though.
If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it unscientific?
Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge.
> There's a difference between something not being instantiated in this universe and being unscientific, though.
Well, science is a particular way of studying what exists. Studying something that doesn't exist is unscientific (of course, you can use science to try to determine IF something exists).
But there are also things that are outside the reach of the methods of science, so they are unscientific in this sense. Questions such as "did some god create the universe" are unscientific because it is simply impossible to apply the methods of science to arrive at an answer to this question.
Similarly, asking "is the universe infinite in size" is unscientific, because it is impossible to apply the methods of science and arrive at a definite answer to this question.
> If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it unscientific?
If it predicts actual infinities exist in certain conditions, than it is not going to be a testable theory in those conditions. It may still be a perfectly workable model, just as GR is perfectly workable despite predicting singularities at the center of black holes. That doesn't mean that the singularities exist, it means that GR breaks down at certain points.
But even if you had a physical theory that relied on something like a Banach-Tarski construction, you could never distinguish between an actual infinity of points, leading to two perfectly solid, perfectly identical spheres; and an arbitrarily large number of points, leading either to two perfectly solid but slightly different-sized spheres; or two identically-sized spheres with small holes.
Of course, without some need to specify the number of points, you would be well positioned to use the infinite variant. But if someone asked you if this means that the sphere really has an infinite number of points, the answer would have to be that you can't be sure.
Aren't claims like this unscientific according to your standard? You will never be able to measure that two electrons have the same charge to infinite decimal precision. You might have a theory that says they should have the same charge, but you won't be able to test that theory to infinite precision either.
Perhaps the problem here is one of mixing intuition (the idea of 'an object') with rigorous physics and mathematics, perhaps this is where I am going a bit wrong.
>Studying something that doesn't exist is unscientific
What about things that could exist, might exist, or even aren't expressly forbidden from existing? These have all been used as perfectly valid reasons for scientific inquiry, historically.
Asking "did some god create the universe" is unscientific by your reasoning so long as it is known that there is no in-universe trace or evidence that it was indeed created by a god. Proving that is proving a negative. I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something. It does certainly feel impossible that we will prove that the universe wasn't created by a god, though. The inquiry is deemed unscientific because we have no reason to go down that pathway, not because the question is fundamentally intractable.
Multiverse theory, on the other hand, would qualify as unscientific by your reasoning. If it were true, the different universes would be fundamentally inaccessible, according to our understanding. The model does not suggest that evidence could even possibly exist, as far as I understand.
A result being untestable doesn't, in my opinion, lead to it being unscientific. We cannot test whether black holes exist, except by looking for them. We cannot test whether wormholes exist, except by looking for them. These are predictions that we cannot "test" except by looking at the universe and seeing what we find, and even then we are not guaranteed a positive result, just because maybe it is the case that our model is correct but there was never the appropriate state of the universe to prove our prediction.
Of course if something was actually infinite, you wouldn't be able to measure it to be so, but if the model (that you have shown to be correct in other case) predicts an actual infinity and you keep counting more and more orders of magnitude, does it not make sense to assume your model is correct? Is that unscientific? Just like we assume that the charge on electrons is constant despite not actually measuring it always everywhere.
> I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something.
That's actually a good point, there could be scientific proof of some intelligent creator in principle. The fact that there is no reason a priori to believe that we will find such a proof is a problem, but I don't think it would be enough to deem the theory unscientific. Otherwise, many actually used theories would be unscientific - for example, there is no scientific reason to expect supersimmetry to exist, but that doesn't make the search for supersimmetry unscientific.
> Multiverse theory, on the other hand, would qualify as unscientific by your reasoning.
Yes, multiverse theory is unscientific by my definition. I don't believe speculation about a multiverse can be considered science in any meaningful sense. Just like simulation theory, it is using science-sounding terminology for idle speculation (though the universe being a simulation could similarly be proven by the same kind of evidence as the intelligent creator idea, to be fair).
> These are predictions that we cannot "test" except by looking at the universe and seeing what we find, and even then we are not guaranteed a positive result
But this is exactly the definition of a test. It's true that you can't prove that something doesn't exist in this way, but saying that something is untestable goes beyond that. An untestable hypothesis is one that by definition doesn't make any predictions about the universe. Multiverse theory is in this bucket - whether you believe it to be true or not, you won't expect to see anything different in the world.
> Of course if something was actually infinite, you wouldn't be able to measure it to be so, but if the model (that you have shown to be correct in other case) predicts an actual infinity and you keep counting more and more orders of magnitude, does it not make sense to assume your model is correct?
Of course it's OK to assume your model is correct, and infinity will likely be the simplest assumption in this case. However, any model that predicts an infinity can be replaced with an equivalent model that makes all the same measurable predictions but replaces the infinity with some arbitrarily large but finite number (or arbitrarily small but not infinitesimal). This second model may well be harder to work with and will contain an extra assumption (an explicit upper bound for the infinite quantity), so I wouldn't advocate for its use. But it would have to be accepted that it is not empirically distinguishable from the infinity based model.
An infinite value is theoretically testable. It simply implies that for however long you make your ruler, the value is larger. That is a prediction. You may not reach a conclusion, as you said, in finite time, but that is still a prediction.
The problem with replacing an infinity in a model with an arbitrarily large number is that, given enough time and a long enough ruler, you'll surpass that number, meaning your model is incorrect. In defence of "science", you're adding an arbitrary number into a model that you expect to be incorrect. That's not how it should work.
If the model says there's a singularity, we don't then say "okay but well that clearly doesn't make sense, so put a limit on the formula that clamps the values to uh 10^45". That is unscientific.
Yes, others have pointed that out and I am in fact conflicted right now.
> The problem with replacing an infinity in a model with an arbitrarily large number is that, given enough time and a long enough ruler, you'll surpass that number, meaning your model is incorrect. In defence of "science", you're adding an arbitrary number into a model that you expect to be incorrect. That's not how it should work.
I'm not against using infinities in scientific practice at all. I'm just pointing out that, when it comes down to it, that infinity is never necessary in the logical sense.
> If the model says there's a singularity, we don't then say "okay but well that clearly doesn't make sense, so put a limit on the formula that clamps the values to uh 10^45". That is unscientific.
Sure, picking some random big number would be unscientific. But saying "the model predicts a singularity or growing to infinity, so we're probably missing some piece of the picture that sets an upper bar" is not unscientific. It is in fact the common practice - just like no one believes that black holes or the early universe had an actual singularity at the center, we normally just believe the models break somewhere at those levels, and more powerful models (quantum gravity) will actually put a cap. Or how we keep saying that we know an upper bound for the possible mass of a photon, but don't actually know that it really is 0, and we keep trying to measure it.
Mathematics has nothing to do with laws of physics. Even if the laws of physics[0] were different, these mathematical[1] truths would remain the same.
[0] Laws of physics don't actually exist. They're shorthand generalizations about features of particulars. The notion of some kind of abstract disembodied "laws" that somehow "govern" everything is absurd.
[1] For clarify: mathematics is a field that studies such things.
Given that, "laws of physics" are certainly describable. We simply write the formulae that tell us what the next state of the dynamical system is. They are ways of delimiting what is physically possible, given the current state of the art.
What is the alternative? That we merely observe rigid patterns that are baked into physical reality? Isn't whatever is 'baked in' more or less a 'law of physics'?
If these are just 'brute facts' are they not then 'laws'? Maybe governance is too strong an word for the correspondence but what is the alternative?
This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.
At best, when you "hand me 7 electrons", you're directing me towards the fat part of 7 probability distributions, so we're back to math again...
[0] https://www.quantamagazine.org/what-is-a-particle-20201112/
Honestly I think that's a continuous claim, and comes down to differences in understanding. I can certainly have 7 of some object, does the 7-ness exist in the collection? Not really, but what about another phenomenon: colour? An object appears blue, and we say it is blue, and the blueness is due to physics, but it's a subjective delineation. A table is a delineation too, the leg is part of the table and the White House is not. In some sense, the table-ness category is just as real as the 7-ness category.
Of course you could just say that all that actually exists is some collection of particles/fields, but then you've abused all the words we're using until they stop being useful.
So, while I can't "point" to an infinite number of things like I can point to 9 things or 3.62 things, I still think it exists.
I'm not sure how well this generalizes to all infinite cardinals, ordinals, or to transfinite induction/construction. It is certainly strange that Cantor's theorem (the cardinality of a set is strictly smaller than that of its power set) implies there are different sizes of "all" implicit in my usage of the word.
What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large.
The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).
This seems incorrect.
You're on the right track.
The Banach-Tarski paradox requires accepting that non-measurable sets[1] exist. A non-measurable set is a set with a an inspecifiable volume. Note: That's non-measurable - not 0. It means you have a quantity of something, whose volume is not 0, but it's also not any other number.
Once I realized that the paradox requires it, all the WTF aspect went away. Of course - if you can accept quantities for which you cannot specify a volume, you can probably accept about anything.
Interpret it as "adding more points will not necessarily increase the volume, no matter how many points you add". There are plenty of measure-0 sets containing as many points as the continuum does.
> This seems incorrect.
It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following:
1. You have a ball.
2. You cut the ball into 5 pieces in a very clever way.
3. You move the pieces around.
4. Now you have two balls, each the same size as the first.
The key, interesting part of this is in step 3, where we only use translations and rotations. Those preserve volume. (By contrast, it's easy to scale a ball of radius 2 to become a ball of radius 3, but that's not a volume-preserving transformation.) The part of the process that doesn't preserve volume is actually step 2, where we cut the ball into pieces. People find it unintuitive that this step doesn't preserve volume.
You can also cut your ball into several pieces and move the pieces around such that you end up with a much larger ball.
You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an uncountably infinite number of translations. The Banach-Tarski theorem says we can do the same thing in only a finite number of translations.
Can you be more specific? Rotations and translations also exist in 2-space. It seems difficult to argue that this difference between 2-space and 3-space is "not a property of 3-space".
Huh? The ball is already composed of infinite points. So in 2a you recognize that the ball exists, and then in 2b you cut it into pieces. But it seems superfluous to mention 2a separately.
If I say a cake is cut into 5 pieces, no person will consider that each piece contains parts from all parts of the cake.
So the paradox breaks down when you start to realize that you are not CUTTING but “choosing some points” and rearranging them. The fact that this rearrangement can be done with Euclidean moves is the surprise.
Not true. If you add uncountably many infinitesimal objects they can add up to noninfinitesimal object, that's how integration works in math, it's pretty confusing cause there's many kinds of infinity and they allow some unintuitive things to happen, but if they didn't worked we couldn't move (see Zeno paradox).
Banach-Tarski is formally correct, you add a finite number of sets with uncountably many points in each so you can get something with volume (depending on how they are positioned).
And yes - a line in math is just a set of points, same with a sphere (but it has 0 volume cause a sphere is just the "skin" without the insides) and a ball (which is what Banach-Tarski talks about). In fact every geometric object is just a set of points.
Addition is defined as an operation with two inputs. You can't add more than two things, unless there is some particular rule that lets you.
If you have finitely many things, then this rule is the associative law: add them pairwise in whatever order, and you are guaranteed to get the same result.
To add infinitely many numbers, you need to talk about limits. Formally, when you say something like
1 - 1/2 + 1/4 - 1/8 + 1/16 - ...
you mean: look at the sum of the first two; then, look at the sum of the first three; then, look at the sum of the first four; and so on -- this sum converges to a limit, which is 2/3.
This sum is "absolutely convergent", which means you get the same result no matter how you order the summands, but some infinite sums change if you reorder things!
With points on the sphere the situation gets even worse, as there is no way to "list them in order". These sets are "uncountable", which means don't even try to sum any function defined on them.
To say approximately the same thing using technical jargon, one has countable additivity for Lebesgue measure on the reals, but uncountable additivity does not hold.
I'll be using spatial dimensions as a conceptual framework to tackle this exact issue in future videos.
That part is easy - for each point on a unitary sphere move it to a point at position 2x ( ie. to a corresponding location on the sphere of the 2 units radius) - you've just doubled the volume, i.e. you've just built a 2 units radius sphere out of the points belonging to 1 unit radius sphere. Banach-Tarski of course more fun and illustrates much more than just volume.
In the paragraph on nonstandard analysis in the Wikipedia page on infinity, it says:
"The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers"
https://en.wikipedia.org/wiki/Infinity
I can't say anything precise or mathematical, but after I read the above, I have an "obvious in hindsight" feeling. If H=inf is different from H + 1, how much different is it? 1/inf or an infinitesimal amount! And an infinitesimal is not nothing.
The quanta article says "You can add or subtract any finite number to infinity and the result is still the same infinity you started with" but this seems like just a dogma for non mathematicians?
They really aren’t connected. The first statement (the positive integers can be partitioned into two sets, each of which has the same size as the original set) follows from the usual axioms of set theory (ZF), while the Banach–Tarski paradox cannot be proven to work without the Axiom of Choice or a similar axiom.
The natural numbers (and therefore Hilbert's Hotel) provide a natural way to say "whatever, just pick one" but we need to invoke the well-ordering theorem (which is equivalent to the Axiom of Choice) make the same "whatever, just pick one" statement about the reals. (and therefore Banach-Tarski)
I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
That's not to say that physics requires infinities, but current models also don't disallow infinity.
Of course, actual infinity is outside the purview of science - there is no way to differentiate between infinity and something too big/small to measure, even in principle. Apparent paradoxes related to infinity, such as Banach-Tarski, don't change this, as they also require infinite precision to realize, making them impossible to test as well - even if a sphere is indeed made up of an infinity of space-time points, and even if we could manipulate those, we wouldn't be able, in finite time, to extract the necessary infinite subsets of points to create the two spheres from one.
---
...It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?"
"No holes?"
"No holes."
"Impossible! There ain't no such thing."
"Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!"
Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms."
"But we have the condition of continuity: We can keep on cutting!"
"No, you said an orange, so I assumed that you meant a real orange."
It's sort of hilarious to see a physics site mention the Banach-Tarski paradox. It is, after all, the most obvious hole poked in the most basic working assumption used by physicists: that space and time are measured with real numbers.
I've seen physicists go to pretty absurd extremes to avoid thinking about the problems this creates. Fixing it properly is not easy: simply dropping the axiom of choice leaves you unable to do useful physics. Getting back to a useful state, making all sets Lebesgue, can only be done with large cardinals:
https://www.jstor.org/stable/1970696
Large cardinals are pretty exotic even by the standards of mathematicians. In many departments they are in fact the domain of logicians. In fact, the existence of certain classes of Woodin cardinals is equivalent to the Axiom of Determinacy (AD), which is the "mathematically respectable" way of investigating logics with infinitary conjunction/disjunction. In fact, AD is precisely the Law of Excluded Middle (A or not-A) for logics with infinitely-long conjunctions.
Quite odd that something so ethereal would be connected to a tangible act like cutting an apple in half.
Assume I have a sphere made of pure iron. I divide the sphere into individual iron atoms. I divide this group of atoms into two groups of atoms. I take each of those groups of atoms and form them into 2 spheres. How is it that these two new spheres are not either less dense or smaller that the original sphere?
You have highly restricted the act of choosing sets of points here. B-T doesn't say that any "division" results in that unintuitive outcome.
Note that points are infinitesimally small and infinitely many, and atoms in your iron sphere are neither.
In reading the comments for the video, I got the sense that this is different and that I was missing something but couldn't come close to guessing what that was.
Looking into it more closely, it turned out to be both trivial and not notably meaningful, like most surprising results involving uncountable infinity. Nothing that affects us involves actual infinities, so infinities are just a convenient approximation that often produces correct-enough answers. Anything infinities imply that seems crazy trivially is.
It is natural to suspect that foundational axioms are somewhere flawed.
TLDR; It is basically the same as proofs that all countable sets have the same cardinality. (TLDR of that: map set of positive integers x to the even numbers by doubling, and the odd numbers by doubling and subtracting 1. Take the union of even and odd and you end up with the set you started with, the positive integers x).
For a circle:
We can identify all the points on a circle as the points p associated with the [x,y] coordinates of the complex numbers p = e^(2.c.i.pi), where 0 <= c < 1. (And . is multiply.)
If we take each of those points p and rotate it by doubling its c, we now have the same points represented by the expression p = e^(2.c.i.pi), where x <= 0 < 2.
So the same number of points, but two passes around the circle, 0 <= c < 1 and 1 <= c < 2. We can move the second set of points in the x positive direction by 2 or more to avoid the overlap.
We have now rearranged points of one circle into two.
For the surface of a sphere:
We simply divide a sphere up into points defined by a stack of circles at real-valued vertical z positions, z <= -1 <= 1. And their real [x,y] points are the real and imaginary parts of each circle e^(2.c.i.pi).circumference(z), where 0 <= c < 1, and circumference(z) is the cos(z).
Again, rotate the points by doubling c, so that they are now located at c, where 0 <= c < 2. There are now two overlapping sphere surfaces. We can move the second in any direction by 2 to avoid the overlap.
Similar generalizations work for including the volume.
Anyone understand why this simpler proof is wrong, or why the more complex proof in the article does something better?
I don't buy the diagonalization proof as anything more than the Pythagoreom Theorom. You have infinite rows, and infinite columns. Infinity is Schrodinger's Cat. Once you check in on the state (nth row by mth column) the only thing you can say about the diagonal number is that is hasn't occurred in the rows up to that point, not beyond, nor in the columns (if n > m).
Ergo, Infinity is a paradox, and only mathematical in the absurd.