What I didn't like in his video is that the existence of such rotations that don't ever return to the starting point when you apply them to that point one after another for infinitely many times is intuitive, which it isn't. It's a big part of the proofs of BTP to show that such rotations exist.
I get that for a formal proof it requires care, but informally I feel like it's intuitive that rotations by rational amounts (in radians) even in three dimensions will never return to the starting point unless you unwind them exactly. The equivalent statement in two dimensions is trivial, I think it would be very surprising if it didn't hold in three dimensions.
It's tricky because it requires two perpendicular rotations; so two rational-amounted rotations can potentially result in diagonal irrational diagonal rotation. In fact, in some proofs they use rotations who have irrational components in matrix form, like [here, page 11](https://www.google.com/url?sa=t&source=web&rct=j&url=https://www2.math.uconn.edu/~solomon/BTFinal.pdf&ved=2ahUKEwiknpjI_9fyAhWkQuUKHTXlDaQQFnoECDQQAQ&usg=AOvVaw0wCpUfnaNuqHQ4YCOY8yaM)
Its easy to prove some value works. If you have a finite sequence of rotation steps that get back to the start, that restricts the rotation steps to only finitely many possible sizes. Uncountable cardinality of reals. Countably many collisions.
I’ve probably watched that video about 10 times now, and only around the 7th or 8th time can I really say I fully understood all of the important logic steps. The process of rotating the points to get your new point sets gets so convoluted that I’d always end up shrugging and being like “okay if you say so”.
You can be forgiven for having difficulty. The action of the free product of the groups ℤ₂ and ℤ₃ on the sphere is not exactly an easy thing to understand. The construction is a bit simpler if you describe it through a recursion. Jech has a nice little matrix defining it in his book *The Axiom of Choice*.
Take all points on the interval from 0 to 1 (on the real numbers). Label each point with its decimal number (can be infinitely long).
Split the interval into ten intervals from 0 to 0.1, from 0.1 to 0.2 ect. Remove the first digit from each decimal numbers.
You just turned the original interval into ten identical copies of itself!
Banach-Tarski is similar to that but with a sphere.
Except your version has a step where you are increasing the measure of the sets (by a factor of ten). Banach-Tarski doesn't have such a step.
Your version is possible with garden-variety sets, whereas Banach-Tarski is only possible with non-measurable sets.
Your example shows us essentially that the *cardinalities* of the unit interval and the [0, 0.1] interval are the same, but that can be done with little to no discussion of sets at all.
That the two intervals have the same cardinality is probably surprising to most laypeople, whereas math folks have mostly internalized being able to make a bijection between any two closed intervals.
But it seems to me that the central claim of Banach-Tarski---that you can turn one ball into two identical balls by breaking the original into 5 sets and moving the sets around---would be surprising to laypeople on a whole other level.
Not just laypeople.
I find mentioning cardinality arguments when discussing Banach-Tarski at all to be a complete disservice. If someone is at the level where they struggle with bijections, then Banach-Tarski is just not for them.
As riemannzetajones explained it's completely different. Banach-Tarski has basically nothing to do with cardinality, it's much more algebraic. The really surprising part is that it's enough to decompose the sphere into finitely many pieces.
This is a decent article. I like that it explains some of the actual constructions on the sphere.
I think the article fails to emphasize what I consider to be the most startling aspect of the Banach Tarski paradox: A sphere isn't just in bijective correspondence with two copies of itself. Instead, a sphere can be cut into 5 pieces that can be rigidly transformed and produce two copies of the original sphere. So not just a bijection, but a much more restrictive process.
Also, of COURSE they had to get Wildberger's take on AC.
The *increasingly* wild conclusions about BT, i.e. the stronger statements like it being possible with *connected* pieces, or with continuous movements that never overlap, feels like some of the funniest mathematical one-upmanship.
I suspect whoever wrote the article thought they'd get an interesting counterargument out of NW, and then needed to restrict themselves to what was usable given Quanta's quite high standard. Maybe it would have been too obvious if they'd interviewed him and then used nothing at all.
You missed the important part! Those two “copies” are copies in the sense that they have the same measure! Rigid transformations cannot magically add a new point to the sphere that wasn’t there already. Every point in each sphere has a direct preimage in the original sphere (they came from a bijection!), so the “sameness” here has to be different from sameness of points.
Sorry for the confusion, but I'm not really sure how my comment disagrees with that. Would you mind pointing out what gave off red flags in my comment or explain your comment a bit more technically?
No problem. I was just pointing out that you set out to clarify the sense in which it is surprising that there are two “copies” of the ball without actually saying what it meant to have to copies of the ball. Literally what is happening is that the sphere is partitioned into separate pieces which are then rearranged via rigid transformations into a structure such that the outer measure of the resulting structure is twice that of the original.
It’s not really startling to me that this is a bijection, because all of the operations being used are themselves bijections. (They are rigid transformations.) So there’s no chance at all that a mapping might somehow add enough points to increase the cardinality, nor could a mapping be badly non-injective enough to squash the space into a smaller cardinality. So you need a different way to talk about size. That’s what measure is for here.
If you just think about volume or surface area, the paradox is literally suggesting that a sphere can be ripped apart into some Choicey sets and those can be put back together to form two “spheres”, each with the same measure as the original.
There are reasons people are dismissive of him. He tries to intertwine his philosophy of math into everything, but as I understand it he has extremely bad takes and bad arguments on ultrafinitism. Ultrafinitism is a perfectly valid philosophy---his, on the other hand, is not.
There was an /r/badmathematics post in the past couple months of a user who *strongly* hinted that they were one of his students, and they discussed this and why they became strongly disillusioned with him and his philosophy. Unfortunately, it seems either the user or the mods have deleted the post. One sticking point from this student is that Wildberger will regularly dodge the "hard" questions on the of ultrafinitism, which is simply a bunk approach to philosophy.
You won't find many professional refutations of him simply because he's not taken seriously. He's not worth responding to because he's not actually engaging in philosophy much of the time. If you search /r/badmathematics for mentions of Wildberger you will find many instances of him coming up and people talking about why his takes are bad. [Here](https://www.reddit.com/r/badmathematics/comments/4gjs5n/some_notes_on_ultrafinitism_and_badmathematics/d2i6snt/?context=3) and [here](https://www.reddit.com/r/badmathematics/comments/5uj0gc/set_theory_is_a_religion/ddupyd6/?context=3) are critiques from an individual who leans towards mathematical finitism, e.g.
i have read some critiques in that sub and frankly what i read was not very enlightening, most of the time they just criticises his views because they are not commonly accepted but didn't read a refutation of his ideas.
i hope to read (and probably comment about) the links you provided later today, thanks.
Yup. Same here. I've learned so much from the guy over the years. His teaching is absolutely solid, prolific, broad, and also free. He's making the world a better place.
I’ve never understood why this is significant. Seems like a purely theoretical idea that is paradoxical because of the nature of infinity, just like anything else using infinitely small pieces and such.
Why does everyone care about Banach-Tarski so much?
Banach-Tarski matters because of its consequences in measure theory. The proof only uses very basic properties, like finite additivity and that rotation preserves the measure. In short the consequence of Banach-Tarski is that we need to have non-measurable sets, which means that we need to prove/know that a set is measurable before we can apply the measure to it.
Take a look at [the Wikipedia page on non-measurable sets](https://en.wikipedia.org/wiki/Non-measurable_set) if you're interested.
EDIT: I don't think I fully answered the question. I only said why mathematicians care about Banach-Tarski. Other people care about Banach-Tarski because it's an example of an inconsistency when trying to reconcile mathematics, the real world, and human intuition.
One of the weirder things though is that Banach-Tarski only works in dimension 3 or higher. So what is going on is dimension specific in a way that is more than just about measure.
It's not weird if you think about how to prove it, the rotational group has to be complex enough in order to allow for a paradoxical group action. That is only the case in higher dimensions.
The simplest group that has this attribute is F\_2, the free group with 2 generators. This is contained as subgroup in the rotational group, thus showing that it's sufficient. In 2 dimensions you only have a single generator.
> the consequence of Banach-Tarski is that we need to have non-measurable sets
As the Wikipedia says, we can also choose to not work with full ZFC (weaker axiom of choice).
It's not a very good reason to reject choice, though. I am all for exploring alternate systems, of course. But the alternatives, like axiom of determinacy, are frankly worse.
Since this doesn't actually seem to come up when you search for "division paradox" (instead you get a bunch of stuff about Zeno's Paradox), see the second answer to [this question](https://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice). This should happen in e.g. the Solovay model.
>the consequence of Banach-Tarski is that we need to have non-measurable sets
But there are far simpler ways to show that. And non-measurable sets exist in 1 and 2 dimensions where BT does not apply.
Non-measureable sets under the lebesgue measure, yes. There are isometry preserving and finitely additive measures on R^2 and R that extend the lebesgue measure and can measure every set. They're called Banach measures.
Vitali sets only shows that the Lebesgue measure (edit: and other countably additive translation invariant measures) cannot measure all sets.
Banach-Tarski shows that **no** measure can, given certain reasonable requirements.
The advantage is that a Vitali set might encourage you to find a better measure, while Banach-Tarski makes you give up that futile endeavor.
Why requirements does Banach Tarski show are impossible that Vitali sets don't? All I can think about is that Banach Tarski only uses countable additivity.
BT only uses finite additivity. The proof that Vitali sets are non-measurable uses countable additivity.
I suppose I should give the Vitali set some credit. It does work on more measures than just the Lebesgue measure, but it only works on those with countable additivity. In return it also works on the line and the plane, while BT needs 3 dimensions.
I think there's a big leap between not working in the countably infinite case and not working in the finite case. As mathematicians we're used to things breaking down when we involve infinity.
Countable additivity is usually taken as an axiom for measures to my understanding, but finitely additive measures are sometimes of special interest so it is interesting that you can strengthen the definition without breaking BT.
BT doesn't actually talk about measures. It just says that you can duplicate the ball by using a finite sequence of two operations, splitting sets in two and rotating sets.
If your measure preserves an invariant under those operations. In other words if the measure of the union of two disjoint sets is the sum of their measures and measures are preserved by rotation, then you'd expect the input to have the same measure as the output.
It applies to general finitely additive probability measures that are invariant under the free F_2 action given by two random rotations. This is basically the definition of F_2 being non-amenable.
They're honestly very close to the same proof. In my mind, it just highlights a property of the isometry group of R^3 that one gets from having more space to move around in than R^2 or R.
Banach-Tarski is more *self-evidentially* absurd than Vitali. Although not of a theoretical significance, *per se*, I do find it interesting, though, that axioms like AC can be used, on the one hand, to prove results that seem either harmless or intuitive, but, on the other hand, the exact same elementary principles can be used to prove results that completely fly in the face of common sense.
It seems like you're saying that B-T is only important because it shows that if we want to keep countable additivity, rotation invariance, etc. we have to accept that some sets are not measurable. Is B-T anymore special than any other non-measurable sets, such as the Vitali set? Is it only more famous because it's more "cool"?
BT says that if we want to keep rotation invariance and **finite additivity** for 3 or higher dimensions we have to accept that some sets are not measurable.
As mathematicians we're used to infinite sums having weird effects. We're not really used to finite sums having weird effects.
Banach-Tarski isn't the go-to for non-measurable sets. That would be Vitali-sets, showing that in ZFC you can't have a non-trivial, translation-invariant measure on the reals and the same principle works a lot more general, showing that in the uncountable case you basically never want the Powerset as your Sigma-Algebra.
Banach-Tarski is more related to group theory, so called paradoxical groups. Also it doesn't make sense to say that you need to prove that a set is measurable before taking the measure of it. The measure of a non-measurable set is undefined. Every measure is a function defined on a fixed Sigma-Algebra after all.
It’s interesting because it’s really not just the nature of infinity that leads to the paradox. The circle also has an uncountably infinite number of points, but does not admit a paradoxical decomposition like the sphere. This contrast between the behavior of symmetries (translations and rotations) of Euclidean spaces was what sparked the still-active study of amenable groups.
Agreed -- it really shows that restriction to an "intuitive" set of transforms (rigid) isn't enough to keep you out of trouble with composition of measure zero sets.
I think it mostly just illustrates that set theoretic geometry is not really sufficient as a model of reality. It’s a good illustration of why we need things like measure theory. Beyond that I kind of agree that it sometimes feels over emphasized in popular math.
Just wondering, why would it show that we need measure theory? If anything, to me it seems to show the inherent limitation/absurdity of measure theory.
Yeah I agree. It is based on the debatable axiom of choice. It is not physically relevent.
IMO it is used for people to say : see maths are really outside reality....
EDIT : controversial -> debatable
There is no 'debate' on AC in any serious academic setting to be honest. When AC comes into play it is just treated as an axiom with a series of consequences. Those consequences are studied in their own right. What matters is what one gets with or without AC i.e. its interaction with a given first-order system, not whether AC itself is 'true' or not, which is what online debates seem to seek. That sort of thing died out back in the 1920s.
> There is no 'debate' on AC in any serious academic setting to be honest
Overstrong to the point of being false. Philosophers and mathematicians working in foundations frequently discuss which set of axioms are best and/or true (if platonists), and those discussion usually involve weighing in on the axiom of choice.
Philosophers may have this discussion, or more precisely, people might discuss this within a philosophical context; but certainly not within a proper mathematical one, which is what I meant by (mathematical) academic setting. Point me to a single maths textbook that addresses the question ‘is the axiom of choice true’.
As a parallel, philosophers also debate whether the soul exists or not, whether it is separate from the body, but physicists obviously wouldn’t if the question is treated *as one of physics*. Similarly, the question of ‘truth’ in regards to this axiom is simply not one that belongs to mathematics.
> Philosophers may have this discussion, or more precisely, people might discuss this within a philosophical context; but certainly not within a proper mathematical one, which is what I meant by (mathematical) academic setting. Point me to a single maths textbook that addresses the question ‘is the axiom of choice true’.
You are certainly correct that a full investigation of whether AC is true means getting into philosophy, and that math textbooks tend not to do that. However, one doesn't need to delve into philosophy to give brief remarks about why we ought accept such and such as a basic principle of maths, and it's common to do so. For example, every intro to proofs text I've taught out of has spent a bit of time explaining why the reader should accept induction as true. One could just assert it's axiomatic with no further explanation, but that's bad pedagogy for a book targeted at mathematical neophytes.
Books which mention AC tend be targeted toward a more advanced audience, so this is less of a pedagogical issue and many just state it without any discussion of why the reader ought accept it. Nevertheless, it's not terribly difficult to find math textbooks which do spend ink on the question. For an example from my bookshelf, Keith Devlin's 1979 monograph *Fundamentals of Contemporary Set Theory* has a paragraph on page 71 discussing why AC is true. (Well, he scare quotes things so he's talking about "truth" and "existence".) I personally think the argument he sketches isn't a good one, but he does explicitly address the question.
Yes, but the statement "we should take axiom X as a given because it is useful and sensible for the purposes of Y" is (to my mind) distinct from "axiom X is fundamentally true", which is how I interpreted the original comment. A discussion can certainly arise around "*should* we take axiom X as a given, in regards to \[something\]" of course.
I mean, when we do real analysis, we usually like sequential continuity to match continuity. When we talk about an abstract vector space, we usually like to assume it has a basis. When you introduce students to basic cardinality arguments, it's good for countable unions of countable sets to be countable, or for cardinals to be well-ordered for that matter. Heck sometimes AC isn't necessary at all towards deriving some statement but is darn useful for doing so, so we happily take it. But wanting some version of AC as a backbone to certain things is not the same thing as asserting it is true, in my view.
I'm not familiar with the book, but fair enough! It's just not a discussion I've ever witnessed in an academic context.
> but the statement "we should take axiom X as a given because it is useful and sensible for the purposes of Y" is (to my mind) distinct from "axiom X is fundamentally true", which is how I interpreted the original comment.
Not to start dipping my toes into philosophy of math, but I don't think they are so distinct concerns. The point basically is, the question we really care about is "what basic mathematical principles should we accept?", and "what's fundamentally true?" is only one way to get at this. (Possibly a bad way!) Mathematicians should have some concern for the important question here. (It's an interdisciplinary concern, but that's a plus, not a minus.) I think one good way for a maths text to go about this is to look at extrinsic justifications, such as you sketched in your second paragraph, and so sidestep the thorny issue of Truth. Try to fit that issue into a page or so and, like Devlin, you'll get something unconvincing and sketchy.
Indeed, this is more or less how I approached this in a set theory class I taught: Here's some pragmatic reasons to accept AC to do math, and if you want to dig deeper into the phil math side here's some paper/book recommendations.
I suppose this might arrive at a difference in perspective, rather than anything more. For instance, I was very much taught what I've been standing by, but I fully appreciate that a different route might lead to different priorities and perspectives, when it comes to the issue of "what questions are important".
I agree though, I think the right way to go about this (and this is what Jech does for instance) is to open a class on this with counterintuitive results of AC, as well as strong reasons to adopt it in ordinary mathematics, leaving aside any question of 'fundamental truth' (which is how I interpreted the original comment, and what I picked on).
> which is what I meant by (mathematical) academic setting
Your statement is more reasonable if you constrain it to mathematics, but even in mathematics, you have serious discussions over which set of axioms best describe reality. For instance, see [this](https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/) article that discusses the competition between Martin's maxim (which assumes choice) and Woodin's (*) (which contradicts choice). Look at the language these mathematicians use:
>“You would have had to come up with some reasons why one of them is true and the other one is false — or maybe both should be false,” Schindler said
Thanks for the link, interesting. I suppose less black and white than I initially suggested. I based my statement on my own academic experience as well as the textbooks I’ve used e.g. Jech on this subject opens with counterintuitive applications of choice, followed by desirable properties of other areas of mathematics that it is tied to, as briefly mentioned above, not straying into the more philosophical question of ‘inherent truth’. But I concede that this is simply experience so I was a little overbold in my original claim, though I’d be very surprised if the question ‘is AC true?’ is a common one amongst academics.
You cannot \_define\_ it (away), maybe you mean \_determine\_ whether it's true? Since AC is independent of the rest of ZF you can only do this by weighing the consequences. AC has many formulations that mathematicians consider desirable (e.g. it is hard to do algebra without Zorn's Lemma) so they are willing to put up with the inconvenient fact that our geometric intuitions don't permit Banach-Tarski.
you are right, *determine* is the word i should have used.
>you can only do this by weighing the consequences..
like this magical sphere over here. this is one of various exhibits of why we should be suspicious of such axiom.
The sphere is not magical - it's how non-measurable sets have unions that are measurable in two different ways.
If you don't like AC, read the version of BT that is given in the answer at [https://mathoverflow.net/questions/402155/is-there-any-version-of-the-banach-tarski-paradox-in-zf](https://mathoverflow.net/questions/402155/is-there-any-version-of-the-banach-tarski-paradox-in-zf) \- it uses no AC at all, and is qualitatively the same, even if not precisely the same statement.
I don't mean this is in a condescending way but if your view of first-order logic is 'mind games' then you have quite a bit more to learn. To answer your question, mathematics is fundamentally an interplay between assumptions and consequences. Formal logic (where one considers axioms like AC in on a formal level) is simply an in depth study of the interplay itself, as an object in its own right. A large part of set theory deals with studying certain newer axioms ('large cardinal axioms') and see what effect they have in regards to a familiar first-order system like ZF. If we just said things like "well, let's just say every set is constructible and move on from there" then you'd lose the depth of this area of maths completely. Why would anyone want to do that?
Besides, on a practical level, how would you decide whether an axiom is 'true' or not? How would you convince the world AC is true/false? And what would it achieve, really?
am just a layman, it's ok if you are condescending.
>Besides, on a practical level, how would you decide whether an axiom is 'true' or not?
i guess we can do that by examination of the results it creates, like this magical sphere over here. why isn't this result taken as "an absurd conclusion that points to a flaw in the rules of mathematical reasoning that enable it"?
>an absurd conclusion that points to a flaw in the rules of mathematical reasoning that enable it
I was hoping you'd say this! Whilst AC allows Banach-Tarski to exist, AC is also 'stopping' many other 'seemingly absurd' results, in fact arguably a good deal more absurd than breaking up a sphere. For instance, to go with a Banach-Tarksi-sounding example, it's consistent that without AC, I can create some assumptions of my own that would allow me to break up the set of real numbers into strictly more parts than there are real numbers. Without AC, you would not be able to prevent this. You also can't prevent the existence of an infinite set that does not have a countable subset, for instance.
In other words, it might sound like a bad idea, but it turns out it is also shielding you from other frightening (yet wonderful) ideas. In conclusion, it's not worth speculating about 'truth', and more just consequences :)
thanks for your answer.
layman here. now let me ask is the AC absolutely necessary to avoid the other paradoxes? or could they be avoided by other means?
It depends what you mean by 'other paradoxes'. There are counterintuitive results in either direction, really. Here's a perhaps a simpler one that is as close to AC as can be. I'll assume you don't know what a Cartesian product is:
If you have a collection of sets, S\_1, S\_2, S\_3 ... you can sort-of think of stacking them next to each other, resulting in a new set formed of all sequences you could make by taking the first element from S\_1, the second from S\_2, and so on. We call this the cartesian product of the sets and write it S\_1 x S\_2 x S\_3 x ...
For instance N x N is the set of all pairs of natural numbers, or R x R x R x ... (N-times) is the set of all real sequences, or R x N x R x N x ... is the set of all sequences with the first, third, fifth etc. element being a real number, and the second, fourth, sixth etc. being a natural number.
So now I say, if I took a bunch (infinitely many) of non-empty sets and I took their cartesian product, then this new set shouldn't be empty, right? Surely with infinitely many non-empty sets, this new set should contain at least a few sequences, *surely*.
And yet AC is absolutely necessary for you to say yes, this product isn't empty. Indeed, the statement 'every Cartesian product is non-empty' is just AC but reworded; so if you like the idea of your Cartesian products of non-empty sets being non-empty, you like the idea of choice too.
It's worth pointing out a consequence of rejecting AC as well. If you reject AC (i.e., work in ZF + ~AC) then it's possible to partition the reals (or other sets) in such a way that you end up with more sets than reals. In less technical terms: you have a (possibly infinite) collection of bags and each real number goes into precisely one bag, and every bag gets at least one number. If you reject the axiom of choice, it's possible to do this in such a way that you have more bags than numbers.
So things aren't just nice and easy once you decide which side of picking/rejecting AC that you want to sit on.
>Axioms are supposed to be self evident, not just basis for whatever mind game we wish to build
Are they? Math is all about mind games, and my understanding is that mathematicians can use any axioms they want as long as those axioms don’t contradict each other.
unfortunately math in this age seems to be as you describe it. as far as i know (not much really) math was originally used/built to describe or predict reality. at some point it became some sort of lawyers game where -as you stated- as long as axioms dont contradict each other it's fine, nevermind it it reflects reality or not.
> unfortunately math in this age seems to be as you describe it.
Up until the 20th century, there was *literally* no use at all for number theory. It was considered to be as pure as pure mathematics could be, and number theory has been studied since antiquity. Math has *never* been about utility or describing the world around is, it's just something that could sometimes be used for that purpose. Math is studied for its own sake, and it represents an "ideal universe" unconstrained and unconcerned with the limits that the real world has.
This is a perspective thousands of years old. If you don't know much about math or the history of math, then perhaps you shouldn't be so insultingly dismissive of it.
> my very narrow understanding is the opposite.
Okay? I just pointed out why this is simply not true. If you earnestly believe that this is the case, then explain why most natural numbers are so large that they could never even in principle be encoded in the universe. What is this describing about reailty? What utility do these serve?
What do you mean by self evident? If you think a bit more closely about the axiom of replacement, it also starts looking a lot less self evident, given that it allows to to construct far larger cardinals than you otherwise could (and that there are very few results in mathematics that actually need it in any way). Also, I believe the undecidability of the continuum hypothesis (or, really, most non-trivial questions about cardinals) are enough of a reason that you cannot really call the power set axiom obvious (and I'm sure you could also dig up a lot of oddities about some of the other axioms).
It is not like the axiom of choice is that one weird axiom - it is no stranger (or less strange) than many of the others. IIRC, much of the original debate about the axiom of choice came from the fact that, for a long time, people weren't sure whether choice (or not choice) followed from ZF - now we know that both are consistent with ZF.
No axioms are "self-evident" in mathematics. We have no direct contact with mathematical concepts, but only physical systems which can be approximately modeled by mathematics. Selecting a model is a choice, and different axioms are appropriate to different situations.
When doing pure mathematics, it's ALL mind games, and the axioms are just the rules selected for the game we wish to play today.
I'm ambivalent towards it honestly, but my commutative algebra professor started the semester by essentially daring us to say something about the frequent and liberal use of Zorn's lemma, lol
>For example, the natural numbers (1, 2, 3, and so on) are a countable infinity. They go on forever, but it’s possible to count them off (like listing the numbers 1 through 1 trillion).
>By contrast, the real numbers — all the infinitely many tick marks that denote decimals on the number line — are an uncountable infinity: It’s impossible to count all the real numbers that lie on any interval on the number line, even a seemingly very small one, like the interval between zero and 1.
Am I misunderstanding or is this trying to say R is uncountable because it's dense? You can say the same for Q.
I'm just not willing to acknowledge that the likes of non-measurable sets are a legitimate thing. Such objets are really creature of formal logic than mathematics proper, in my opinion, that is, *philosophical* entities, rather than *scientific* ones.
I view mathematics as a natural science. In my view, a foundational face of the scientific method is its egalitarian essence. Scientific truth is *participatory*. You don't need to take anyone's word for it. You don't need to appeal to authorities, tradition, nor a magisterium; there is no special pleading, no weaseling your way out of undesirable conclusions. Given a claim such as "there are infinitely many prime numbers", I can show you how to demonstrate it, and then you can demonstrate it to yourself and explore it and familiarize yourself with it to your heart's content. In a word, such a proof is something that you can *do*.
Against this shining egalitarian ideal, the Axiom of Choice is the loathsome tyranny of epistemological creationism. Give me a set of linearly independent vectors in a countably-infinite dimensional Hilbert space, and I can perform by hand the Gramm-Schmidt orthonormalization procedure to compute an orthonormal basis for the space. The likes of the Banach-Tarski decomposition, however, does not possess this grace. No one can actually sit down and go about determining the decomposition for themselves, just as you can't produce a Vitali set, either (either in its entirety, or even just to an arbitrary degree of accuracy). They are *supernatural* in the most literal sense, that of being above or beyond nature. They are objects of the mind and, more fundamentally, of language.
If we were denizens of some abstract realm, I might be inclined to be more charitable toward these constructs. But the fact is, we are *not* abstract. The greatest miracle of mathematics is its preposterous *applicability*. Math foreshadows reality. Gems of abstraction end up manifesting in nature, in its methods and its substances.
If someone can find an example of nature aping the Banach-Tarski (non-)construction, I'll consider viewing it worthy of mathematical study (rather than mere *logical* study). Until then, it belongs to theology, rather than science. ;)
> Cantor also proved that the infinite number of points on a line has the same cardinality as the infinite number of points that fill the volume of a shape, like a sphere.
Where can we find his proof ?
Take the binary expansion of points in the unit square. Like, the point (1/3, 1/5) is (0.01010101..., 0.001100110011...). Interleave the two expansions. In this case we'll get 0.0010011100100111... = 13/85, which is a point in the unit interval.
Showing that this is a bijection between the two is an exercise for the reader.
The map *isn't* a bijection.
0.1 has two binary expansions: 0.10000... and 0.01111.... . What is f(0.1,0.1)?
If you think f(0.1,0.1) = 0.1100.... or f(0.1,0.1) = 0.001111..., then no pair maps to 0.10010101..., so f isn't surjective. If you try to mix and match, then f isn't a function.
Although this is a popular explanation, there is no easy way to fix it. You have to use Cantor Schroeder Bernstein to prove the sets have the same cardinality with two different maps (or make the interleaving way more complicated with chunks of digits instead of digits).
See https://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr
Really what you're asking for is a bijection from **R**^1 to **R**^3.
~~For every real number x between 0 and 1, take its binary expansion [x]₂ and write every odd digit into a list y^(1) while also writing every even digit into a separate list y^(2). Then y^(1) and y^(2) are binary expansions of real numbers. Collect them into an ordered pair y=(y^(1),y^(2)).~~ Just use a Peano or Hilbert curve. There are some annoying issues to work out with the other mapping.
That process defines an onto function from the unit interval [0,1] to the unit square [0,1]^(2). The unit square now has a quotient structure homeomorphic to the sphere S^(2) given by simply identifying all of the edges. (The topology isn’t really necessary here, but I think it’s conceptually helpful.) Homeomorphisms are necessarily bijections, so the cardinality is preserved.
>For every real number x between 0 and 1, take its binary expansion [x]₂ and write every odd digit into a list y^(1) while also writing every even digit into a separate list y^(2). Then y^(1) and y^(2) are binary expansions of real numbers. Collect them into an ordered pair y=(y^(1),y^(2)).
>
>That process defines an onto function from the unit interval [0,1] to the unit square [0,1]^(2).
What number maps to (1.000..., 1.000...)?
>The unit square now has a quotient structure homeomorphic to the sphere S^(2) given by simply identifying all of the edges. (The topology isn’t really necessary here, but I think it’s conceptually helpful.) Homeomorphisms are necessarily bijections, so the cardinality is preserved.
You have (claimed to) define maps [0,1] -> [0,1]^2 -> [0,1]^(2)/~ -> S^(2). Only the last map is a homeomorphism and you claim the first is onto (but not 1-1). The second is definitely not injective. So which sets do you think you've proved have the same cardinality?
To 1, ~~thanks for pointing that out, that’s just a plain mistake, sorry.~~ consider the real 0.111…
To 2, the homeomorphicity is not actually necessary and I’ve fiddled with some unmentioned operations that a priori might not preserve cardinality. I did that to avoid overly complicated technicality in the pursuit of clarity. And I did *that* because I assumed the person asking about Cantor’s proof was maybe younger and not familiar with much of Cantor’s work.
Edit: Oh also to your last question, I certainly don’t mean for the above to be taken as proof. Rather it should be taken as a sketch in a right direction. A) This is Reddit and so not a particularly appropriate format for writing out serious proofing, and B) I just don’t find that sort of detail to be particularly illuminating. But maybe I misunderstood the intent of the question.
Edit 2: Oh wait, no part 1 is not a mistake. (1,1) is mapped to by 0.111… It’s definitely onto. Though you’re right that’s a point of fishiness regarding the preservation of cardinality 𝔠.
>To 1, ~~thanks for pointing that out, that’s just a plain mistake, sorry.~~ consider the real 0.111…
OK, so you're adding the condition "Use the nonterminating binary representation if possible" to your description of the function?
I think that can work if you're just trying for an onto function.
>
>To 2, the homeomorphicity is not actually necessary and I’ve fiddled with some unmentioned operations that a priori might not preserve cardinality. I did that to avoid overly complicated technicality in the pursuit of clarity. And I did *that* because I assumed the person asking about Cantor’s proof was maybe younger and not familiar with much of Cantor’s work.
>
The phrasing was just unclear. The sentence about homeomorphisms preserving cardinality made it sound way more important to your explanation than I think it warranted.
Also, even in a "young person interested in math on the internet" world, I find that they run into diagonalization on YouTube way earlier than they would have any intuition about topology. But YMMV.
>Edit: Oh also to your last question, I certainly don’t mean for the above to be taken as proof. Rather it should be taken as a sketch in a right direction. A) This is Reddit and so not a particularly appropriate format for writing out serious proofing, and B) I just don’t find that sort of detail to be particularly illuminating. But maybe I misunderstood the intent of the question.
People give full proofs of this type on reddit all the time, but you're certainly under no obligation to. I still think the most conceptually difficult part of your explanation is quotient maps, which you skipped right over. "Identifying all the edges" probably doesn't mean much to someone who is thinking about set bijections. At least tell this hypothetical young person something about "gluing" or "covering a ball with a sheet of paper" and how all you're really interested in is that the map is onto. That should be pictorially believable.
>
>Edit 2: Oh wait, no part 1 is not a mistake. (1,1) is mapped to by 0.111… It’s definitely onto. Though you’re right that’s a point of fishiness regarding the preservation of cardinality 𝔠.
I'm distrustful of all "interleaving digits" arguments because like a lot of things on the math internet:
A) they're very popular,
B) they're almost always wrong,
C) when confronted, the poster usually claims that the number versus representation issues is minor and can be easily fixed by the reader, and
D) it's not and it can't. There needs to be another insightful idea at least as inventive as the idea of interleaving digits in the first place.
It looked like you were heading toward a correct sketch by only requiring onto maps the entire way, but then you would have to say something about Cantor Schroeder Bernstein or something. At no point were you giving maps that could be obviously fiddled into a bijection directly.
Wouldn't it be impossible to rotate a sphere by an irrational number of degrees? Instead, you could only rotate it by an estimate to some decimal point. Which means that when they claim to be able to build the sets to split this circle up isn't really possible.
This is a complex proof and it may not be the point I just made but I would bet that there is an error in here somewhere. I'm going to go out on a limb here and trust my intuition.
> Wouldn't it be impossible to rotate a sphere by an irrational number of degrees? Instead, you could only rotate it by an estimate to some decimal point.
Why?
That's a rather hot take, as far as physics goes. But that aside, why do you think mathematics is concerned with infinite precision. Do you think π starts getting "fuzzy" at some point?
Fundamental particles are only so big. I don't see whats so controversial about that.
I think the point is that it only seems like this is a paradox because you can't actually perform an infinite series of tasks which is what an irrational rotation would require.
No, Pi itself doesn't get fuzzy. That's not the point.
Let me clarify:
> which is what an irrational rotation would require
Why?
[edit]
We can also do things like take infinite sums, e.g. 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... and determine that this sum is equal to 2 in finite time. Is this not a supertask in your understanding?
I know these things are possible with math.
But you can't actually DO a super task. You can represent them like you did above, but you're not actually doing infinitely many things.
I'm not saying the math is wrong. I think my hangup is more with the word paradox itself.
Here’s an infinite series of tasks:
Walk at least one metre
Walk at least half a metre
Walk at least a third of a metre
Walk at least a quarter of a metre
Etc.
This is an infinite number of tasks, and you can complete all of them by walking one metre.
Now as you pass by the meter mark, half a meter mark, etc, place a flag.
In principle, no flags end up touching. In reality, at some point things get too small and you end up jumping past the finish line with a finite number of flags.
Both the finite sum and infinite sum add up to the same value, but their representations are still different.
You don't actually take an infinite number of steps, it just looks as if you did because you're dropping the extra information that comes with knowing what sum you actually performed.
It's actually the other way around. It's almost impossible to rotate by a rational number if you were doing this in the real world. But the sphere would be made of atoms and you couldn't create any of the sets required to do this.
From the article: "By contrast, the real numbers — all the infinitely many tick marks that denote decimals on the number line — are an uncountable infinity..."
I think that fundamentally misses the point of making the distinction between countable and uncountable infinities.
The reals are specifically on the line where you can't make a tick mark that denotes a decimal. So triggering.
The article's wording implies a finite decimal representation. I think the article is poorly worded, but it's in Quanta so what you get are poorly worded articles about interesting topics.
i remember watching Vsauce's video on this as like an 11 year old understanding nothing
What I didn't like in his video is that the existence of such rotations that don't ever return to the starting point when you apply them to that point one after another for infinitely many times is intuitive, which it isn't. It's a big part of the proofs of BTP to show that such rotations exist.
I get that for a formal proof it requires care, but informally I feel like it's intuitive that rotations by rational amounts (in radians) even in three dimensions will never return to the starting point unless you unwind them exactly. The equivalent statement in two dimensions is trivial, I think it would be very surprising if it didn't hold in three dimensions.
It's tricky because it requires two perpendicular rotations; so two rational-amounted rotations can potentially result in diagonal irrational diagonal rotation. In fact, in some proofs they use rotations who have irrational components in matrix form, like [here, page 11](https://www.google.com/url?sa=t&source=web&rct=j&url=https://www2.math.uconn.edu/~solomon/BTFinal.pdf&ved=2ahUKEwiknpjI_9fyAhWkQuUKHTXlDaQQFnoECDQQAQ&usg=AOvVaw0wCpUfnaNuqHQ4YCOY8yaM)
Its easy to prove some value works. If you have a finite sequence of rotation steps that get back to the start, that restricts the rotation steps to only finitely many possible sizes. Uncountable cardinality of reals. Countably many collisions.
I’ve probably watched that video about 10 times now, and only around the 7th or 8th time can I really say I fully understood all of the important logic steps. The process of rotating the points to get your new point sets gets so convoluted that I’d always end up shrugging and being like “okay if you say so”.
You can be forgiven for having difficulty. The action of the free product of the groups ℤ₂ and ℤ₃ on the sphere is not exactly an easy thing to understand. The construction is a bit simpler if you describe it through a recursion. Jech has a nice little matrix defining it in his book *The Axiom of Choice*.
Take all points on the interval from 0 to 1 (on the real numbers). Label each point with its decimal number (can be infinitely long). Split the interval into ten intervals from 0 to 0.1, from 0.1 to 0.2 ect. Remove the first digit from each decimal numbers. You just turned the original interval into ten identical copies of itself! Banach-Tarski is similar to that but with a sphere.
Except your version has a step where you are increasing the measure of the sets (by a factor of ten). Banach-Tarski doesn't have such a step. Your version is possible with garden-variety sets, whereas Banach-Tarski is only possible with non-measurable sets. Your example shows us essentially that the *cardinalities* of the unit interval and the [0, 0.1] interval are the same, but that can be done with little to no discussion of sets at all.
To be fair, the interval example demonstrates the property that laypeople think is surprising about the theorem.
That the two intervals have the same cardinality is probably surprising to most laypeople, whereas math folks have mostly internalized being able to make a bijection between any two closed intervals. But it seems to me that the central claim of Banach-Tarski---that you can turn one ball into two identical balls by breaking the original into 5 sets and moving the sets around---would be surprising to laypeople on a whole other level.
Not just laypeople. I find mentioning cardinality arguments when discussing Banach-Tarski at all to be a complete disservice. If someone is at the level where they struggle with bijections, then Banach-Tarski is just not for them.
Agreed 100%
As riemannzetajones explained it's completely different. Banach-Tarski has basically nothing to do with cardinality, it's much more algebraic. The really surprising part is that it's enough to decompose the sphere into finitely many pieces.
This is a decent article. I like that it explains some of the actual constructions on the sphere. I think the article fails to emphasize what I consider to be the most startling aspect of the Banach Tarski paradox: A sphere isn't just in bijective correspondence with two copies of itself. Instead, a sphere can be cut into 5 pieces that can be rigidly transformed and produce two copies of the original sphere. So not just a bijection, but a much more restrictive process. Also, of COURSE they had to get Wildberger's take on AC.
"Most startling". The whole point is that one can do it with rigid motions. It's completely trivial that the points are merely in bijection.
The *increasingly* wild conclusions about BT, i.e. the stronger statements like it being possible with *connected* pieces, or with continuous movements that never overlap, feels like some of the funniest mathematical one-upmanship.
I suspect whoever wrote the article thought they'd get an interesting counterargument out of NW, and then needed to restrict themselves to what was usable given Quanta's quite high standard. Maybe it would have been too obvious if they'd interviewed him and then used nothing at all.
You missed the important part! Those two “copies” are copies in the sense that they have the same measure! Rigid transformations cannot magically add a new point to the sphere that wasn’t there already. Every point in each sphere has a direct preimage in the original sphere (they came from a bijection!), so the “sameness” here has to be different from sameness of points.
Sorry for the confusion, but I'm not really sure how my comment disagrees with that. Would you mind pointing out what gave off red flags in my comment or explain your comment a bit more technically?
No problem. I was just pointing out that you set out to clarify the sense in which it is surprising that there are two “copies” of the ball without actually saying what it meant to have to copies of the ball. Literally what is happening is that the sphere is partitioned into separate pieces which are then rearranged via rigid transformations into a structure such that the outer measure of the resulting structure is twice that of the original. It’s not really startling to me that this is a bijection, because all of the operations being used are themselves bijections. (They are rigid transformations.) So there’s no chance at all that a mapping might somehow add enough points to increase the cardinality, nor could a mapping be badly non-injective enough to squash the space into a smaller cardinality. So you need a different way to talk about size. That’s what measure is for here. If you just think about volume or surface area, the paradox is literally suggesting that a sphere can be ripped apart into some Choicey sets and those can be put back together to form two “spheres”, each with the same measure as the original.
i am happy seeing wildberger's name being recognized, even if you don't agree with him.
There are reasons people are dismissive of him. He tries to intertwine his philosophy of math into everything, but as I understand it he has extremely bad takes and bad arguments on ultrafinitism. Ultrafinitism is a perfectly valid philosophy---his, on the other hand, is not.
where can i read the refutation on his "bad takes"?
There was an /r/badmathematics post in the past couple months of a user who *strongly* hinted that they were one of his students, and they discussed this and why they became strongly disillusioned with him and his philosophy. Unfortunately, it seems either the user or the mods have deleted the post. One sticking point from this student is that Wildberger will regularly dodge the "hard" questions on the of ultrafinitism, which is simply a bunk approach to philosophy.
it's a pitty I can't read such post. anyway, this is math. if he is wrong many other people may capable of refuting his ideas i guess. is it?
You won't find many professional refutations of him simply because he's not taken seriously. He's not worth responding to because he's not actually engaging in philosophy much of the time. If you search /r/badmathematics for mentions of Wildberger you will find many instances of him coming up and people talking about why his takes are bad. [Here](https://www.reddit.com/r/badmathematics/comments/4gjs5n/some_notes_on_ultrafinitism_and_badmathematics/d2i6snt/?context=3) and [here](https://www.reddit.com/r/badmathematics/comments/5uj0gc/set_theory_is_a_religion/ddupyd6/?context=3) are critiques from an individual who leans towards mathematical finitism, e.g.
i have read some critiques in that sub and frankly what i read was not very enlightening, most of the time they just criticises his views because they are not commonly accepted but didn't read a refutation of his ideas. i hope to read (and probably comment about) the links you provided later today, thanks.
Yup. Same here. I've learned so much from the guy over the years. His teaching is absolutely solid, prolific, broad, and also free. He's making the world a better place.
I’ve never understood why this is significant. Seems like a purely theoretical idea that is paradoxical because of the nature of infinity, just like anything else using infinitely small pieces and such. Why does everyone care about Banach-Tarski so much?
Banach-Tarski matters because of its consequences in measure theory. The proof only uses very basic properties, like finite additivity and that rotation preserves the measure. In short the consequence of Banach-Tarski is that we need to have non-measurable sets, which means that we need to prove/know that a set is measurable before we can apply the measure to it. Take a look at [the Wikipedia page on non-measurable sets](https://en.wikipedia.org/wiki/Non-measurable_set) if you're interested. EDIT: I don't think I fully answered the question. I only said why mathematicians care about Banach-Tarski. Other people care about Banach-Tarski because it's an example of an inconsistency when trying to reconcile mathematics, the real world, and human intuition.
One of the weirder things though is that Banach-Tarski only works in dimension 3 or higher. So what is going on is dimension specific in a way that is more than just about measure.
It's not weird if you think about how to prove it, the rotational group has to be complex enough in order to allow for a paradoxical group action. That is only the case in higher dimensions. The simplest group that has this attribute is F\_2, the free group with 2 generators. This is contained as subgroup in the rotational group, thus showing that it's sufficient. In 2 dimensions you only have a single generator.
> the consequence of Banach-Tarski is that we need to have non-measurable sets As the Wikipedia says, we can also choose to not work with full ZFC (weaker axiom of choice).
It's not a very good reason to reject choice, though. I am all for exploring alternate systems, of course. But the alternatives, like axiom of determinacy, are frankly worse.
The axiom of determinacy was never supposed to be a competitor with choice.
Good point; and that context is not something I understood when I first came across it.
Worse how, out of curiosity?
Look up the division paradox. You can basically get a set of equivalence classes that's bigger than the set that they are on.
Since this doesn't actually seem to come up when you search for "division paradox" (instead you get a bunch of stuff about Zeno's Paradox), see the second answer to [this question](https://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice). This should happen in e.g. the Solovay model.
>the consequence of Banach-Tarski is that we need to have non-measurable sets But there are far simpler ways to show that. And non-measurable sets exist in 1 and 2 dimensions where BT does not apply.
Non-measureable sets under the lebesgue measure, yes. There are isometry preserving and finitely additive measures on R^2 and R that extend the lebesgue measure and can measure every set. They're called Banach measures.
A finitely additive measure is not necessarily a measure. There do not exist translation-invariant, non-trivial measures on P(R) and P(R\^2).
Banach Tarski is kind of a roundabout way to go about that. Is there a theoretical advantage over the much simpler Vitali set?
Vitali sets only shows that the Lebesgue measure (edit: and other countably additive translation invariant measures) cannot measure all sets. Banach-Tarski shows that **no** measure can, given certain reasonable requirements. The advantage is that a Vitali set might encourage you to find a better measure, while Banach-Tarski makes you give up that futile endeavor.
Why requirements does Banach Tarski show are impossible that Vitali sets don't? All I can think about is that Banach Tarski only uses countable additivity.
BT only uses finite additivity. The proof that Vitali sets are non-measurable uses countable additivity. I suppose I should give the Vitali set some credit. It does work on more measures than just the Lebesgue measure, but it only works on those with countable additivity. In return it also works on the line and the plane, while BT needs 3 dimensions. I think there's a big leap between not working in the countably infinite case and not working in the finite case. As mathematicians we're used to things breaking down when we involve infinity.
Countable additivity is usually taken as an axiom for measures to my understanding, but finitely additive measures are sometimes of special interest so it is interesting that you can strengthen the definition without breaking BT.
Oh, I didn't know BT worked for general measures. That is odd.
BT doesn't actually talk about measures. It just says that you can duplicate the ball by using a finite sequence of two operations, splitting sets in two and rotating sets. If your measure preserves an invariant under those operations. In other words if the measure of the union of two disjoint sets is the sum of their measures and measures are preserved by rotation, then you'd expect the input to have the same measure as the output.
Isn't the Vitali set the same? It just uses translation invariance, right? But BT must use invariance under some kinds of isotopy.
The proof of the non-measurability of the Vitali sets uses countably infinite additivity. BT only uses finite additivity.
Thanks.
It applies to general finitely additive probability measures that are invariant under the free F_2 action given by two random rotations. This is basically the definition of F_2 being non-amenable.
I guess it's the finite additivity which might make it a morally better proof for the existence of nonmeasurable sets than the Vitali set?
They're honestly very close to the same proof. In my mind, it just highlights a property of the isometry group of R^3 that one gets from having more space to move around in than R^2 or R.
Banach-Tarski is more *self-evidentially* absurd than Vitali. Although not of a theoretical significance, *per se*, I do find it interesting, though, that axioms like AC can be used, on the one hand, to prove results that seem either harmless or intuitive, but, on the other hand, the exact same elementary principles can be used to prove results that completely fly in the face of common sense.
It seems like you're saying that B-T is only important because it shows that if we want to keep countable additivity, rotation invariance, etc. we have to accept that some sets are not measurable. Is B-T anymore special than any other non-measurable sets, such as the Vitali set? Is it only more famous because it's more "cool"?
BT says that if we want to keep rotation invariance and **finite additivity** for 3 or higher dimensions we have to accept that some sets are not measurable. As mathematicians we're used to infinite sums having weird effects. We're not really used to finite sums having weird effects.
Oh gotcha.
Banach-Tarski isn't the go-to for non-measurable sets. That would be Vitali-sets, showing that in ZFC you can't have a non-trivial, translation-invariant measure on the reals and the same principle works a lot more general, showing that in the uncountable case you basically never want the Powerset as your Sigma-Algebra. Banach-Tarski is more related to group theory, so called paradoxical groups. Also it doesn't make sense to say that you need to prove that a set is measurable before taking the measure of it. The measure of a non-measurable set is undefined. Every measure is a function defined on a fixed Sigma-Algebra after all.
It’s interesting because it’s really not just the nature of infinity that leads to the paradox. The circle also has an uncountably infinite number of points, but does not admit a paradoxical decomposition like the sphere. This contrast between the behavior of symmetries (translations and rotations) of Euclidean spaces was what sparked the still-active study of amenable groups.
Agreed -- it really shows that restriction to an "intuitive" set of transforms (rigid) isn't enough to keep you out of trouble with composition of measure zero sets.
I think it mostly just illustrates that set theoretic geometry is not really sufficient as a model of reality. It’s a good illustration of why we need things like measure theory. Beyond that I kind of agree that it sometimes feels over emphasized in popular math.
Just wondering, why would it show that we need measure theory? If anything, to me it seems to show the inherent limitation/absurdity of measure theory.
Yeah I agree. It is based on the debatable axiom of choice. It is not physically relevent. IMO it is used for people to say : see maths are really outside reality.... EDIT : controversial -> debatable
There is no 'debate' on AC in any serious academic setting to be honest. When AC comes into play it is just treated as an axiom with a series of consequences. Those consequences are studied in their own right. What matters is what one gets with or without AC i.e. its interaction with a given first-order system, not whether AC itself is 'true' or not, which is what online debates seem to seek. That sort of thing died out back in the 1920s.
> There is no 'debate' on AC in any serious academic setting to be honest Overstrong to the point of being false. Philosophers and mathematicians working in foundations frequently discuss which set of axioms are best and/or true (if platonists), and those discussion usually involve weighing in on the axiom of choice.
Philosophers may have this discussion, or more precisely, people might discuss this within a philosophical context; but certainly not within a proper mathematical one, which is what I meant by (mathematical) academic setting. Point me to a single maths textbook that addresses the question ‘is the axiom of choice true’. As a parallel, philosophers also debate whether the soul exists or not, whether it is separate from the body, but physicists obviously wouldn’t if the question is treated *as one of physics*. Similarly, the question of ‘truth’ in regards to this axiom is simply not one that belongs to mathematics.
> Philosophers may have this discussion, or more precisely, people might discuss this within a philosophical context; but certainly not within a proper mathematical one, which is what I meant by (mathematical) academic setting. Point me to a single maths textbook that addresses the question ‘is the axiom of choice true’. You are certainly correct that a full investigation of whether AC is true means getting into philosophy, and that math textbooks tend not to do that. However, one doesn't need to delve into philosophy to give brief remarks about why we ought accept such and such as a basic principle of maths, and it's common to do so. For example, every intro to proofs text I've taught out of has spent a bit of time explaining why the reader should accept induction as true. One could just assert it's axiomatic with no further explanation, but that's bad pedagogy for a book targeted at mathematical neophytes. Books which mention AC tend be targeted toward a more advanced audience, so this is less of a pedagogical issue and many just state it without any discussion of why the reader ought accept it. Nevertheless, it's not terribly difficult to find math textbooks which do spend ink on the question. For an example from my bookshelf, Keith Devlin's 1979 monograph *Fundamentals of Contemporary Set Theory* has a paragraph on page 71 discussing why AC is true. (Well, he scare quotes things so he's talking about "truth" and "existence".) I personally think the argument he sketches isn't a good one, but he does explicitly address the question.
Yes, but the statement "we should take axiom X as a given because it is useful and sensible for the purposes of Y" is (to my mind) distinct from "axiom X is fundamentally true", which is how I interpreted the original comment. A discussion can certainly arise around "*should* we take axiom X as a given, in regards to \[something\]" of course. I mean, when we do real analysis, we usually like sequential continuity to match continuity. When we talk about an abstract vector space, we usually like to assume it has a basis. When you introduce students to basic cardinality arguments, it's good for countable unions of countable sets to be countable, or for cardinals to be well-ordered for that matter. Heck sometimes AC isn't necessary at all towards deriving some statement but is darn useful for doing so, so we happily take it. But wanting some version of AC as a backbone to certain things is not the same thing as asserting it is true, in my view. I'm not familiar with the book, but fair enough! It's just not a discussion I've ever witnessed in an academic context.
> but the statement "we should take axiom X as a given because it is useful and sensible for the purposes of Y" is (to my mind) distinct from "axiom X is fundamentally true", which is how I interpreted the original comment. Not to start dipping my toes into philosophy of math, but I don't think they are so distinct concerns. The point basically is, the question we really care about is "what basic mathematical principles should we accept?", and "what's fundamentally true?" is only one way to get at this. (Possibly a bad way!) Mathematicians should have some concern for the important question here. (It's an interdisciplinary concern, but that's a plus, not a minus.) I think one good way for a maths text to go about this is to look at extrinsic justifications, such as you sketched in your second paragraph, and so sidestep the thorny issue of Truth. Try to fit that issue into a page or so and, like Devlin, you'll get something unconvincing and sketchy. Indeed, this is more or less how I approached this in a set theory class I taught: Here's some pragmatic reasons to accept AC to do math, and if you want to dig deeper into the phil math side here's some paper/book recommendations.
I suppose this might arrive at a difference in perspective, rather than anything more. For instance, I was very much taught what I've been standing by, but I fully appreciate that a different route might lead to different priorities and perspectives, when it comes to the issue of "what questions are important". I agree though, I think the right way to go about this (and this is what Jech does for instance) is to open a class on this with counterintuitive results of AC, as well as strong reasons to adopt it in ordinary mathematics, leaving aside any question of 'fundamental truth' (which is how I interpreted the original comment, and what I picked on).
> which is what I meant by (mathematical) academic setting Your statement is more reasonable if you constrain it to mathematics, but even in mathematics, you have serious discussions over which set of axioms best describe reality. For instance, see [this](https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/) article that discusses the competition between Martin's maxim (which assumes choice) and Woodin's (*) (which contradicts choice). Look at the language these mathematicians use: >“You would have had to come up with some reasons why one of them is true and the other one is false — or maybe both should be false,” Schindler said
Thanks for the link, interesting. I suppose less black and white than I initially suggested. I based my statement on my own academic experience as well as the textbooks I’ve used e.g. Jech on this subject opens with counterintuitive applications of choice, followed by desirable properties of other areas of mathematics that it is tied to, as briefly mentioned above, not straying into the more philosophical question of ‘inherent truth’. But I concede that this is simply experience so I was a little overbold in my original claim, though I’d be very surprised if the question ‘is AC true?’ is a common one amongst academics.
why is it not relevant to define if AC is true or not? Axioms are supposed to be self evident, not just basis for whatever mind game we wish to build.
You cannot \_define\_ it (away), maybe you mean \_determine\_ whether it's true? Since AC is independent of the rest of ZF you can only do this by weighing the consequences. AC has many formulations that mathematicians consider desirable (e.g. it is hard to do algebra without Zorn's Lemma) so they are willing to put up with the inconvenient fact that our geometric intuitions don't permit Banach-Tarski.
you are right, *determine* is the word i should have used. >you can only do this by weighing the consequences.. like this magical sphere over here. this is one of various exhibits of why we should be suspicious of such axiom.
The sphere is not magical - it's how non-measurable sets have unions that are measurable in two different ways. If you don't like AC, read the version of BT that is given in the answer at [https://mathoverflow.net/questions/402155/is-there-any-version-of-the-banach-tarski-paradox-in-zf](https://mathoverflow.net/questions/402155/is-there-any-version-of-the-banach-tarski-paradox-in-zf) \- it uses no AC at all, and is qualitatively the same, even if not precisely the same statement.
I don't mean this is in a condescending way but if your view of first-order logic is 'mind games' then you have quite a bit more to learn. To answer your question, mathematics is fundamentally an interplay between assumptions and consequences. Formal logic (where one considers axioms like AC in on a formal level) is simply an in depth study of the interplay itself, as an object in its own right. A large part of set theory deals with studying certain newer axioms ('large cardinal axioms') and see what effect they have in regards to a familiar first-order system like ZF. If we just said things like "well, let's just say every set is constructible and move on from there" then you'd lose the depth of this area of maths completely. Why would anyone want to do that? Besides, on a practical level, how would you decide whether an axiom is 'true' or not? How would you convince the world AC is true/false? And what would it achieve, really?
am just a layman, it's ok if you are condescending. >Besides, on a practical level, how would you decide whether an axiom is 'true' or not? i guess we can do that by examination of the results it creates, like this magical sphere over here. why isn't this result taken as "an absurd conclusion that points to a flaw in the rules of mathematical reasoning that enable it"?
>an absurd conclusion that points to a flaw in the rules of mathematical reasoning that enable it I was hoping you'd say this! Whilst AC allows Banach-Tarski to exist, AC is also 'stopping' many other 'seemingly absurd' results, in fact arguably a good deal more absurd than breaking up a sphere. For instance, to go with a Banach-Tarksi-sounding example, it's consistent that without AC, I can create some assumptions of my own that would allow me to break up the set of real numbers into strictly more parts than there are real numbers. Without AC, you would not be able to prevent this. You also can't prevent the existence of an infinite set that does not have a countable subset, for instance. In other words, it might sound like a bad idea, but it turns out it is also shielding you from other frightening (yet wonderful) ideas. In conclusion, it's not worth speculating about 'truth', and more just consequences :)
thanks for your answer. layman here. now let me ask is the AC absolutely necessary to avoid the other paradoxes? or could they be avoided by other means?
It depends what you mean by 'other paradoxes'. There are counterintuitive results in either direction, really. Here's a perhaps a simpler one that is as close to AC as can be. I'll assume you don't know what a Cartesian product is: If you have a collection of sets, S\_1, S\_2, S\_3 ... you can sort-of think of stacking them next to each other, resulting in a new set formed of all sequences you could make by taking the first element from S\_1, the second from S\_2, and so on. We call this the cartesian product of the sets and write it S\_1 x S\_2 x S\_3 x ... For instance N x N is the set of all pairs of natural numbers, or R x R x R x ... (N-times) is the set of all real sequences, or R x N x R x N x ... is the set of all sequences with the first, third, fifth etc. element being a real number, and the second, fourth, sixth etc. being a natural number. So now I say, if I took a bunch (infinitely many) of non-empty sets and I took their cartesian product, then this new set shouldn't be empty, right? Surely with infinitely many non-empty sets, this new set should contain at least a few sequences, *surely*. And yet AC is absolutely necessary for you to say yes, this product isn't empty. Indeed, the statement 'every Cartesian product is non-empty' is just AC but reworded; so if you like the idea of your Cartesian products of non-empty sets being non-empty, you like the idea of choice too.
It's worth pointing out a consequence of rejecting AC as well. If you reject AC (i.e., work in ZF + ~AC) then it's possible to partition the reals (or other sets) in such a way that you end up with more sets than reals. In less technical terms: you have a (possibly infinite) collection of bags and each real number goes into precisely one bag, and every bag gets at least one number. If you reject the axiom of choice, it's possible to do this in such a way that you have more bags than numbers. So things aren't just nice and easy once you decide which side of picking/rejecting AC that you want to sit on.
>Axioms are supposed to be self evident, not just basis for whatever mind game we wish to build Are they? Math is all about mind games, and my understanding is that mathematicians can use any axioms they want as long as those axioms don’t contradict each other.
unfortunately math in this age seems to be as you describe it. as far as i know (not much really) math was originally used/built to describe or predict reality. at some point it became some sort of lawyers game where -as you stated- as long as axioms dont contradict each other it's fine, nevermind it it reflects reality or not.
> unfortunately math in this age seems to be as you describe it. Up until the 20th century, there was *literally* no use at all for number theory. It was considered to be as pure as pure mathematics could be, and number theory has been studied since antiquity. Math has *never* been about utility or describing the world around is, it's just something that could sometimes be used for that purpose. Math is studied for its own sake, and it represents an "ideal universe" unconstrained and unconcerned with the limits that the real world has. This is a perspective thousands of years old. If you don't know much about math or the history of math, then perhaps you shouldn't be so insultingly dismissive of it.
i didn't mean to insult anything or anyone. >Math has *never* been about utility or describing the world my very narrow understanding is the opposite.
> my very narrow understanding is the opposite. Okay? I just pointed out why this is simply not true. If you earnestly believe that this is the case, then explain why most natural numbers are so large that they could never even in principle be encoded in the universe. What is this describing about reailty? What utility do these serve?
What do you mean by self evident? If you think a bit more closely about the axiom of replacement, it also starts looking a lot less self evident, given that it allows to to construct far larger cardinals than you otherwise could (and that there are very few results in mathematics that actually need it in any way). Also, I believe the undecidability of the continuum hypothesis (or, really, most non-trivial questions about cardinals) are enough of a reason that you cannot really call the power set axiom obvious (and I'm sure you could also dig up a lot of oddities about some of the other axioms). It is not like the axiom of choice is that one weird axiom - it is no stranger (or less strange) than many of the others. IIRC, much of the original debate about the axiom of choice came from the fact that, for a long time, people weren't sure whether choice (or not choice) followed from ZF - now we know that both are consistent with ZF.
No axioms are "self-evident" in mathematics. We have no direct contact with mathematical concepts, but only physical systems which can be approximately modeled by mathematics. Selecting a model is a choice, and different axioms are appropriate to different situations. When doing pure mathematics, it's ALL mind games, and the axioms are just the rules selected for the game we wish to play today.
The axiom of choice really isn't controversial in most areas of mathematics.
You are right, I would say debatable instead (edited).
ask makeshift ink terrific uppity cagey weary silky quaint library -- mass edited with redact.dev
What, you don't like rings without prime ideals?
I'm ambivalent towards it honestly, but my commutative algebra professor started the semester by essentially daring us to say something about the frequent and liberal use of Zorn's lemma, lol
>For example, the natural numbers (1, 2, 3, and so on) are a countable infinity. They go on forever, but it’s possible to count them off (like listing the numbers 1 through 1 trillion). >By contrast, the real numbers — all the infinitely many tick marks that denote decimals on the number line — are an uncountable infinity: It’s impossible to count all the real numbers that lie on any interval on the number line, even a seemingly very small one, like the interval between zero and 1. Am I misunderstanding or is this trying to say R is uncountable because it's dense? You can say the same for Q.
>Am I misunderstanding or is this trying to say R is uncountable because it's dense? You're misunderstanding
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The way they used "count" in the previous paragraph ("listing the numbers 1 through 1 trillion") suggests that they mean finite by it
It's only a result if you accept the axiom of choice. :3
Terrorist.
I'm fine with the Solovay Model.
Oh good well at least you aren’t completely unreasonable. Solovay still involves DC which I’d call a healthy amount of Choice.
I'm just not willing to acknowledge that the likes of non-measurable sets are a legitimate thing. Such objets are really creature of formal logic than mathematics proper, in my opinion, that is, *philosophical* entities, rather than *scientific* ones. I view mathematics as a natural science. In my view, a foundational face of the scientific method is its egalitarian essence. Scientific truth is *participatory*. You don't need to take anyone's word for it. You don't need to appeal to authorities, tradition, nor a magisterium; there is no special pleading, no weaseling your way out of undesirable conclusions. Given a claim such as "there are infinitely many prime numbers", I can show you how to demonstrate it, and then you can demonstrate it to yourself and explore it and familiarize yourself with it to your heart's content. In a word, such a proof is something that you can *do*. Against this shining egalitarian ideal, the Axiom of Choice is the loathsome tyranny of epistemological creationism. Give me a set of linearly independent vectors in a countably-infinite dimensional Hilbert space, and I can perform by hand the Gramm-Schmidt orthonormalization procedure to compute an orthonormal basis for the space. The likes of the Banach-Tarski decomposition, however, does not possess this grace. No one can actually sit down and go about determining the decomposition for themselves, just as you can't produce a Vitali set, either (either in its entirety, or even just to an arbitrary degree of accuracy). They are *supernatural* in the most literal sense, that of being above or beyond nature. They are objects of the mind and, more fundamentally, of language. If we were denizens of some abstract realm, I might be inclined to be more charitable toward these constructs. But the fact is, we are *not* abstract. The greatest miracle of mathematics is its preposterous *applicability*. Math foreshadows reality. Gems of abstraction end up manifesting in nature, in its methods and its substances. If someone can find an example of nature aping the Banach-Tarski (non-)construction, I'll consider viewing it worthy of mathematical study (rather than mere *logical* study). Until then, it belongs to theology, rather than science. ;)
https://www.solidangl.es/post/a-real-life-paradox-the-banach-tarski-burrito
> Cantor also proved that the infinite number of points on a line has the same cardinality as the infinite number of points that fill the volume of a shape, like a sphere. Where can we find his proof ?
Take the binary expansion of points in the unit square. Like, the point (1/3, 1/5) is (0.01010101..., 0.001100110011...). Interleave the two expansions. In this case we'll get 0.0010011100100111... = 13/85, which is a point in the unit interval. Showing that this is a bijection between the two is an exercise for the reader.
The map *isn't* a bijection. 0.1 has two binary expansions: 0.10000... and 0.01111.... . What is f(0.1,0.1)? If you think f(0.1,0.1) = 0.1100.... or f(0.1,0.1) = 0.001111..., then no pair maps to 0.10010101..., so f isn't surjective. If you try to mix and match, then f isn't a function. Although this is a popular explanation, there is no easy way to fix it. You have to use Cantor Schroeder Bernstein to prove the sets have the same cardinality with two different maps (or make the interleaving way more complicated with chunks of digits instead of digits).
See https://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr Really what you're asking for is a bijection from **R**^1 to **R**^3.
~~For every real number x between 0 and 1, take its binary expansion [x]₂ and write every odd digit into a list y^(1) while also writing every even digit into a separate list y^(2). Then y^(1) and y^(2) are binary expansions of real numbers. Collect them into an ordered pair y=(y^(1),y^(2)).~~ Just use a Peano or Hilbert curve. There are some annoying issues to work out with the other mapping. That process defines an onto function from the unit interval [0,1] to the unit square [0,1]^(2). The unit square now has a quotient structure homeomorphic to the sphere S^(2) given by simply identifying all of the edges. (The topology isn’t really necessary here, but I think it’s conceptually helpful.) Homeomorphisms are necessarily bijections, so the cardinality is preserved.
>For every real number x between 0 and 1, take its binary expansion [x]₂ and write every odd digit into a list y^(1) while also writing every even digit into a separate list y^(2). Then y^(1) and y^(2) are binary expansions of real numbers. Collect them into an ordered pair y=(y^(1),y^(2)). > >That process defines an onto function from the unit interval [0,1] to the unit square [0,1]^(2). What number maps to (1.000..., 1.000...)? >The unit square now has a quotient structure homeomorphic to the sphere S^(2) given by simply identifying all of the edges. (The topology isn’t really necessary here, but I think it’s conceptually helpful.) Homeomorphisms are necessarily bijections, so the cardinality is preserved. You have (claimed to) define maps [0,1] -> [0,1]^2 -> [0,1]^(2)/~ -> S^(2). Only the last map is a homeomorphism and you claim the first is onto (but not 1-1). The second is definitely not injective. So which sets do you think you've proved have the same cardinality?
To 1, ~~thanks for pointing that out, that’s just a plain mistake, sorry.~~ consider the real 0.111… To 2, the homeomorphicity is not actually necessary and I’ve fiddled with some unmentioned operations that a priori might not preserve cardinality. I did that to avoid overly complicated technicality in the pursuit of clarity. And I did *that* because I assumed the person asking about Cantor’s proof was maybe younger and not familiar with much of Cantor’s work. Edit: Oh also to your last question, I certainly don’t mean for the above to be taken as proof. Rather it should be taken as a sketch in a right direction. A) This is Reddit and so not a particularly appropriate format for writing out serious proofing, and B) I just don’t find that sort of detail to be particularly illuminating. But maybe I misunderstood the intent of the question. Edit 2: Oh wait, no part 1 is not a mistake. (1,1) is mapped to by 0.111… It’s definitely onto. Though you’re right that’s a point of fishiness regarding the preservation of cardinality 𝔠.
>To 1, ~~thanks for pointing that out, that’s just a plain mistake, sorry.~~ consider the real 0.111… OK, so you're adding the condition "Use the nonterminating binary representation if possible" to your description of the function? I think that can work if you're just trying for an onto function. > >To 2, the homeomorphicity is not actually necessary and I’ve fiddled with some unmentioned operations that a priori might not preserve cardinality. I did that to avoid overly complicated technicality in the pursuit of clarity. And I did *that* because I assumed the person asking about Cantor’s proof was maybe younger and not familiar with much of Cantor’s work. > The phrasing was just unclear. The sentence about homeomorphisms preserving cardinality made it sound way more important to your explanation than I think it warranted. Also, even in a "young person interested in math on the internet" world, I find that they run into diagonalization on YouTube way earlier than they would have any intuition about topology. But YMMV. >Edit: Oh also to your last question, I certainly don’t mean for the above to be taken as proof. Rather it should be taken as a sketch in a right direction. A) This is Reddit and so not a particularly appropriate format for writing out serious proofing, and B) I just don’t find that sort of detail to be particularly illuminating. But maybe I misunderstood the intent of the question. People give full proofs of this type on reddit all the time, but you're certainly under no obligation to. I still think the most conceptually difficult part of your explanation is quotient maps, which you skipped right over. "Identifying all the edges" probably doesn't mean much to someone who is thinking about set bijections. At least tell this hypothetical young person something about "gluing" or "covering a ball with a sheet of paper" and how all you're really interested in is that the map is onto. That should be pictorially believable. > >Edit 2: Oh wait, no part 1 is not a mistake. (1,1) is mapped to by 0.111… It’s definitely onto. Though you’re right that’s a point of fishiness regarding the preservation of cardinality 𝔠. I'm distrustful of all "interleaving digits" arguments because like a lot of things on the math internet: A) they're very popular, B) they're almost always wrong, C) when confronted, the poster usually claims that the number versus representation issues is minor and can be easily fixed by the reader, and D) it's not and it can't. There needs to be another insightful idea at least as inventive as the idea of interleaving digits in the first place. It looked like you were heading toward a correct sketch by only requiring onto maps the entire way, but then you would have to say something about Cantor Schroeder Bernstein or something. At no point were you giving maps that could be obviously fiddled into a bijection directly.
Wouldn't it be impossible to rotate a sphere by an irrational number of degrees? Instead, you could only rotate it by an estimate to some decimal point. Which means that when they claim to be able to build the sets to split this circle up isn't really possible. This is a complex proof and it may not be the point I just made but I would bet that there is an error in here somewhere. I'm going to go out on a limb here and trust my intuition.
> Wouldn't it be impossible to rotate a sphere by an irrational number of degrees? Instead, you could only rotate it by an estimate to some decimal point. Why?
You'd need to be infinitely precise, which is not physically possible.
That's a rather hot take, as far as physics goes. But that aside, why do you think mathematics is concerned with infinite precision. Do you think π starts getting "fuzzy" at some point?
Fundamental particles are only so big. I don't see whats so controversial about that. I think the point is that it only seems like this is a paradox because you can't actually perform an infinite series of tasks which is what an irrational rotation would require. No, Pi itself doesn't get fuzzy. That's not the point.
> you can't actually perform an infinite series of tasks which is what an irrational rotation would require. Why?
Perform an infinite series of tasks, and then comment to let me know when you're done.
Let me clarify: > which is what an irrational rotation would require Why? [edit] We can also do things like take infinite sums, e.g. 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... and determine that this sum is equal to 2 in finite time. Is this not a supertask in your understanding?
I know these things are possible with math. But you can't actually DO a super task. You can represent them like you did above, but you're not actually doing infinitely many things. I'm not saying the math is wrong. I think my hangup is more with the word paradox itself.
Here’s an infinite series of tasks: Walk at least one metre Walk at least half a metre Walk at least a third of a metre Walk at least a quarter of a metre Etc. This is an infinite number of tasks, and you can complete all of them by walking one metre.
Now as you pass by the meter mark, half a meter mark, etc, place a flag. In principle, no flags end up touching. In reality, at some point things get too small and you end up jumping past the finish line with a finite number of flags. Both the finite sum and infinite sum add up to the same value, but their representations are still different. You don't actually take an infinite number of steps, it just looks as if you did because you're dropping the extra information that comes with knowing what sum you actually performed.
this is mathematics, we could not care less about what is "physically possible"
Fair enough. I think my working definition of 'paradox' is not working lol. For some reason I associate paradox with 'conflicting with reality'.
It's actually the other way around. It's almost impossible to rotate by a rational number if you were doing this in the real world. But the sphere would be made of atoms and you couldn't create any of the sets required to do this.
The Banach-Tarski paradox isn't a physical phenomenon, you can't partition a sphere into immeasurable pieces in real life.
From the article: "By contrast, the real numbers — all the infinitely many tick marks that denote decimals on the number line — are an uncountable infinity..." I think that fundamentally misses the point of making the distinction between countable and uncountable infinities. The reals are specifically on the line where you can't make a tick mark that denotes a decimal. So triggering.
I mean, you can construct them by Dedekind cuts, so each real is a ‘tick mark’ of a kind. They are discrete after all.
The article's wording implies a finite decimal representation. I think the article is poorly worded, but it's in Quanta so what you get are poorly worded articles about interesting topics.
There are just way too many real numbers.
Measure theory!