This video is somewhat misleading. I appreciate the attempt at making Banach-Tarski accessible to a general audience, but it dwells on the wrong aspects of what makes Banach-Tarski interesting, making the construction look more like a magic trick with sleight-of-hand. I wish the video had at least mentioned the Axiom of Choice somewhere, as that is fundamentally what Banach-Tarski is about.
The sleight-of-hand comes in around 14:30 into the video, where we are told to create the sequence for an "uncountably infinite number of starting points." That's exactly the point where the construction is non-constructive, and the infamous Axiom of Choice is used. There is no construction - in the sense of constructive mathematics - that can achieve what is described at this point.
Banach-Tarski is not generally regarded as some deep fact about mathematics, a point the video mistakenly belabors. Rather, it is a consequence about particular axiomatizations of set theory which admit the Axiom of Choice. Banach-Tarski is only valid with the Axiom of Choice, and in fact that is the main interest in the paradox.
In my personal opinion, the Banach-Tarski paradox isn't much more enlightening than the simpler construction of the Vitali set (assuming the Axiom of Choice), which is a non-measurable set of real numbers (with Lebesgue measure, i.e., length).
Another part of the video I find misleading has to do with the hyper-dictionary, where he describes the hyper-dictionary by putting some parts of the dictionary "after" other parts which are infinitely long.
The putative applications of Banach-Tarski to physics are ridiculous. Uncountable sets are fundamentally unphysical. The Axiom of Choice serves mainly as a convenience to mathematicians when either a proof avoiding the Axiom of Choice would be more complicated, or so that mathematicians can state properties of objects which are set-theoretically larger than anything that can be relevant to physics anyways.
I think he did: you don't need real numbers to formulate classical physics. Everything measurable has finite precision so you can always get away without postulating that your limits actually converge to something.
Of course, the reality is that then you would wind up with awkward limit-taking machinery in your answers. Real numbers encapsulate that complexity so you might as well use them to simplify both the notation and manipulation of limits. But you don't need to.
Yep, this is what I meant to allude to, and you've worded it much better than I could have.
Perhaps a nice way to say it is that the mathematical objects necessary for physics that I can think of are separable (such as the real numbers). Basically, whenever you have uncountable sets, they come along with some topological structure which must be handled continuously.
There's even an argument that separability is sufficient for most of mathematics let alone physics (http://arxiv.org/pdf/math/0509245.pdf) but you're going to have to work very hard to persuade me that if we're going to describe any sets as physical then uncountable ones are less physical than countable ones.
The set of real numbers is uncountable. There are infinitely many members.
The set of natural numbers is countable. There are infinitely many members.
It would be better to say that infinite sets are fundamentally unphysical, since there are no actual infinities. Talk about uncountable versus countable is tangential.
I have to commend Michael Stevens (VSauce) for the courage to take on Banach-Tarski at all. I was surprised that someone could make it that accessible inside of 20 minutes.
I do agree that a "there is mathematical dispute [++] over whether this operation is valid" disclaimer could have accompanied the part where AoC was used.
[++] Side note: "mathematical dispute" doesn't mean what many non-mathematicians think it does. AoC isn't as "controversial" as some make it out to be. There is no disagreement over whether it's "true" or "false" in any Platonic sense. Rather, there is nearly universal acknowledgement that a valid mathematical system exists with AoC and that a valid mathematical system exists without it... and that both have useful properties. Mathematical dispute over an axiom means that a valid mathematics exists with or without the axiom-- not necessarily that mathematicians hold strong and divergent opinions on whether to include it.
As for the physical interpretations, he's right in that nothing that we know about physics rules out a Banach-Tarski-like behavior at the subatomic level. That said, it's obviously impossible to Banach-Tarski an orange or a ball of gold, since we'd have to literally split every atom. A macroscopic Banach-Tarski event is almost certainly impossible and would be, even with some unforeseen physical capability that made it possible, extremely expensive in terms of energy due to mass-energy equivalence.
His initial explanation of uncountable infinity isn't exactly correct, and I'm afraid it will give people the wrong idea. He says that the real numbers are uncountable, because even between 0 and 1 on the number line there are an infinite number of real numbers. But that is also true of the rational numbers, which are countable! After all, what is the smallest rational number larger than 0?
at each step you list the numbers where the numerator and denominator are <= x. For example, if x = 2, we can count 1, -1, 2, -2, 1/2, and -1/2. Obviously it is an infinite list, but you can list them.
This video is much better than I expected. If you have 25 minutes you can see a friendly presentation of the proof.
[The connection with particle physics at the end I a little too much.]
An important detail is that this video is made by Vsauce. He usually has good videos, but sometimes the connections between the parts are too farfetched.
I consider uncountability an artefact of a flawed approach to mathematics. I'd recommend looking into constructive mathematics based on intuitionistic logic. All fruitful insights based on other mathematics can be proven by it, from all of it's results we can easily extract methods and it is much less mystic. I think it is much more fun, too.
I'm aware of constructivism, but what is "intuitionistic logic?" I am under the impression that the real numbers (and their uncountability) are generally accepted by constructivists.
I would take a look at Andrej Bauer's excellent talk "Stages of Accepting Constructive Mathematics"[0]. Essentially, constructive mathematics is mathematics done without the law of excluded middle ("for all propositions P, either P or not P"). Importantly, it does not necessitate that you deny the law of excluded middle.
Intuitionism (or intuitionistic mathematics) is sometimes used to mean the same thing as constructivism and sometimes refers to Brouwer's[1] practice of denying excluded middle by introducing an axiom along the lines of "all functions are computable" or "all functions are continuous" (which from some perspectives is almost the same thing).
Intuitionistic logic is nearly classical logic. That is, we start by assigning statements one of the two values ‘True‘ and ’False’. While truth in the classical sense is abstract, it is concretized in intuitionistic logic with the meaning of ’we can prove it’. We know that there are statements that can neither be proved nor disproved, so we cannot make use of the law of excluded middle "a statement is true or it is not". Add that we are consistent, that is "not (a and not a)" for all a, and then negation must not be the inverse of itself, because we'd be able to proof the law excluded middle otherwise. You see, it is basically classical logic with some minor adaptions to take into account what we’ve learned.
I understand under "constructive reals" the computable reals, and there are only countably many of them.
It seems misleading to say that intuitionistic logic assigns statements to one of the two values, True or False. There's no symmetry between the notions of truth and falsity as there is in boolean logic.
You need to be very, very careful when talking about the size of the constructive reals. If you are working within constructive mathematics, if you describe the real numbers as setoids of Cauchy sequences with Cauchy-equivalence as the equivalence relation, then there are uncountably many real numbers. From a meta-theoretic perspective, it is obvious (as we are working in constructive mathematics) that every real number is in some sense "computable".
Your notion that there are countably many constructive reals probably comes from definitions of the constructive reals from within classical set theory, wherein you must internalize some notion of what it means for a real number to be constructible, and so you are, in a sense, working meta-theoretically. Then it is no surprise that the computable real numbers are countable. After all, Skolem's paradox says that, meta-theoretically, we could have countable models of the classical real numbers as well.
Additionally, meta-theoretically, we see that our definition of real numbers in constructive mathematics will also have a countable model when seen from the outside.
if you describe the real numbers as setoids of Cauchy sequences
with Cauchy-equivalence as the equivalence relation, then there
are uncountably many real numbers.
I start with computable sequences (or — if want — with turing machines), define natural numbers based on them, go to the rational numbers, define real numbers as a setoid of cauchy-sequences (note: all these sequences will be computable by the way we constructed them) of rational numbers, and end up with countably many. So, it depends on what you start your constructive world with. I see no way to start with something uncountable.
You talk about meta-theory and models. I never saw the reason to complicate things with that. Care to elaborate?
The reals one defines thee reals constructively in essentially the same way as one does classically. When the parent commenter says the reals in constructive mathematics are uncountable, he means that inside of a constructive system one can prove the statement
There does not exist a bijection between the natural numbers and the real numbers.
which is true. You are free to interpret constructive logic in various ways. If you interpret it as the logic of computable things, in which case this statement becomes interpreted as something like
There is no computable bijection between the natural numbers and the computable reals.
If you interpret it into classical logic, it becomes (now in the classical sense)
There is no bijection between the natural numbers and the reals.
ihm's response to you I think did a great job explaining what I meant to say, but let me elaborate further.
Just like you might define computable real numbers, you may similarly describe the "definable" real numbers, that is, those numbers which are uniquely specified by logical statements. You will inevitably find that the definable real numbers, like the computable reals, appear to be countable.
So the fact that something is computable, per se, isn't exactly what makes the real numbers countable rather than uncountable.
I use the vocabulary of meta-theory and models, because the practice of defining either the countable reals or the definable reals within a formal system looks a whole lot like defining a formal system within itself (i.e., metatheory). So the fact that the computable reals and the definable reals appear countable, is, at least to me, much like the statement that there are countable models of your favorite formal system.
HN doesn't allow comments with multiple parents, so I needed to choose one to reply to. ihm, please consider this comment also a reply to yours.
The more I talk about constructivism the less I feel it's adequate to label what I'm talking about, even though my approach to mathematics is completely constructive. Please don't treat every word I use as totally definite.
Mathematics is the science of formal reasoning. I call the rules of reasoning, that is the rules of manipulation of statements including their lexicon, syntax and semantic, a logic. I see intuitionistic logic as the most intuitive, universal and fruitful logic, in short: the logic.
To reason, we need something to reason about. Said somethings should be respresentable, so we can communicate about it. Let me call the things we reason about objects. What do we use to represent objects? Symbols, pictures, sounds, smells and many more things we have senses for. We transmit objects as a literal, or as an algorithm ("a turing machine") of how to recreate the object. All turing machines can be encoded with finite bit arrays.
It turns out we are able to make digital computers recreate symbols and pictures and sounds to a fidelity such that we can't notice any difference compared to the original. I see no fundamental difficulty besides physical limitations to recreate other literals. All literals in a computer can be encoded with finite bit arrays.
Finally, I conclude that finite bit arrays are the most general objects we can reason about; they have rich semantics by the possibility to interpret them as turing machines. What we end up with looks a lot like common programming languages.
Based on intuitionistic logic, finite bit arrays, and the ability to interpret them as turing machines, we can follow the canonical construction of the reals. In this world, every object is a finite bit array, plus some annotated interpretation ("a type"), which is just notation for a bigger finite array. There is simply no place for non-countability.
I long for a software that lets me mix logic and usual programming seamlessly, without unnecessary like meta-theories, meta-logic. Coq is near, but the programming is too unnatural.
One can reason about the logic of a Rust programm, discuss it with others, and even the compiler understands some of the meaning, but richer intuitionistic logic statements about the output or behaviour of the code are sadly beyond the compiler.
I'm about to start applying for my graduate studies of mathematics. Do you have any recommendations which university to apply to to research in the area I talked about?
I think there is no difference between constructive and classical logic here? The proof of "the powerset of the natural is uncountable" (or equivalently, expanding out the definition of uncountability, "there does not exist a bijection between N and P(N)") is constructive, and will hold just as well in intuinistic logic. You can't get away from uncountable sets.
On the other hand, you could prove a metatheoretic result, constructing a model which only contains sets which are forced to exist by some finite formula (and therefore the model is countable). But that you can do in ordinary classical set theory also (the Löwenheim-Skolem theorem, Gödel's constructible universe).
So is your intuitionist discipline a subset of constructivism? I thought that constructivists generally accept the existence and uncountability of the real numbers.
I don’t feel safe enough with the vocabulary, so please let me use my own words to describe it. Construcive mathematics implies for me to construct all objects you talk about on the things you already declared (for example, recursion becomes a notational shortcut). So, we need to start with something. One simple approach would be finite bit arrays. The most general approach I see, and the one I take, is to start with computable sequences of bits. Consequently, everything based on that is countable.
They're usually used synonymously. Yes, Cantor's diagonal argument can still be used to show the uncountability of real numbers in a constructive setting.
Vsauce makes a good point that we just aren't made to intuitively understand this type of stuff. Recognizing this fact, in my opinion, is a key driver in helping to wrap our minds around these concepts. It's important to understand that these concepts are valid in both our visible world and the hidden quantum world, the main difference is scale.
For example, I could never wrap my head around the fact that electrons can have multiple paths/histories simultaneously when travelling. The same is true of a baseball thrown in the air, the only difference is that on the visible scale that we are used to the chance of that baseball taking a different path/history is so small that it will never happen.
I actually think it's very possible to have an "intuitive understanding" of Banach Tarski (I would say I have one, but perhaps we disagree on what is mean by such an understanding).
My "intuitive understanding" of this comes via an intuitive understanding of a paradoxical decomposition of the free group and its Cayley graph, which is flashed briefly in the video here[0] but sadly not discussed at length.
Infinity is a concept, not a number. Confusing the two is what gets you into trouble. And, unfortunately, it is easy to confuse them because in mathematical notation, infinity is often used in place of a number.
There are plenty of senses in which infinity IS a number -- or rather, many numbers. See e.g. the Wikipedia articles on cardinal numbers, ordinal numbers, hyperreal numbers, and surreal numbers.
It's his passion for the material that always keeps me watching. Listening to someone explain a topic that they find deeply intriguing is a pleasure. Reminds me of Feynman.
from layperson's intuition - isnt this just a trick performed by extracting the extra elements from an infinity ? sort of the opposite to losing information through common scaling by zero ?
2 = 1 because (2)0 = (1)0
a matter of defining rules and staying consistent to them. the paradox arises from expecting a contrived model to manifest in physical reality
The sleight-of-hand comes in around 14:30 into the video, where we are told to create the sequence for an "uncountably infinite number of starting points." That's exactly the point where the construction is non-constructive, and the infamous Axiom of Choice is used. There is no construction - in the sense of constructive mathematics - that can achieve what is described at this point.
Banach-Tarski is not generally regarded as some deep fact about mathematics, a point the video mistakenly belabors. Rather, it is a consequence about particular axiomatizations of set theory which admit the Axiom of Choice. Banach-Tarski is only valid with the Axiom of Choice, and in fact that is the main interest in the paradox.
In my personal opinion, the Banach-Tarski paradox isn't much more enlightening than the simpler construction of the Vitali set (assuming the Axiom of Choice), which is a non-measurable set of real numbers (with Lebesgue measure, i.e., length).
Another part of the video I find misleading has to do with the hyper-dictionary, where he describes the hyper-dictionary by putting some parts of the dictionary "after" other parts which are infinitely long.
The putative applications of Banach-Tarski to physics are ridiculous. Uncountable sets are fundamentally unphysical. The Axiom of Choice serves mainly as a convenience to mathematicians when either a proof avoiding the Axiom of Choice would be more complicated, or so that mathematicians can state properties of objects which are set-theoretically larger than anything that can be relevant to physics anyways.