He went into a lot of the technical detail, but the most important base principle is ♾️ x 2 = ♾️ The practical-sounding description of “cutting up a ball and putting the pieces together into two balls” disguises the fact that at its core, this is like Hilbert’s Infinite Hotel
I first encountered the Banach-Tarski paradox in my subject for mathematical proof. When we discussed certain set-theoretical concepts, we naturally covered the Axiom of Choice. Our teacher introduced us to the Banach-Tarski paradox and promised we would eventually learn its proof as we attended higher mathematical classes. I needed to learn concepts from mathematical analysis and topology to actually understand the proof.
@@Schmidtelpunkt If you read the proofs of the Banach-Tarski paradox, you'll need extensive knowledge on group theory, set theory, analysis, and linear algebra. The Axiom of Choice is usually introduced in foundational math courses (usually where principles of mathematical proof is introduced).
@@Schmidtelpunkt This proof is longer, but is more detailed. Plus, I love how the author manages to explain some of the concepts. www.diva-portal.org/smash/get/diva2:1672461/FULLTEXT01.pdf
@@Schmidtelpunkt The Axiom of Choice has such incredibly diverse and apparently formulations I once wrote a paper which merely listed these various formulation, with a discussion of how bizarrely unconnected these are I'll take a stab at it by calling it the mathematician's get out of jail free card for proof-writing - with the realization that it's an adequate conceptualization
It makes it possible to deconstruct the (measurable) ball into non measurable sets that, if reassembled into a (measurable) set, happen to have a different volume. I don't think explaining that would fit into a Ted video aimed at a general audience.
if you have no mathematical knowledge besides highschool, it would probably take you around a year of studies that are necessary to understand the proof
Yes, because the tiny fraction of the audience that would understand it would be all made of people who already understood it. I guess one of the goals here is to inspire people, in particular young people, to seek that kind of knowledge. But the technical parts of it require years of intense study, of course. Anyway, understand the basics of mathematics, as in what is an axiom and what is a theorem, is much more accessible and was indeed covered here.
@@julianbruns7459 By measurable, you mean in the Lebesgue integration sense? Dear lord, that was my waterloo This from a person whose area of expertise is the pinnacle of an introductory graduate level course in abstract algebra Namely Galois Theory
@@נועםדוד-י8ד I'll postulate that it would take anyone - under those condition - a heck of a lot longer Unless they were extremely gifted in math In which case, they'd have known that already and therefore have much more than a basic understanding
The Banach Tarski paradox is that its theoretically possible for there to be an infinite number of options and the same option to be picked every time. If you have a bag of M&Ms and put each M&M into a different box in a room with infinity number of boxes its theoretically possible for an all knowing being to choose those same M&Ms each time.
Wait, I actually was able to follow through with this! Basically, math itself is pretty abstract but it becomes concrete once we apply it in a practical situation (a.k.a. reality). And there are alot of alternate "truths", I guess, that would lead to different realities. Cool stuff.
Not an infinite number (at 0:15), a finite number. From Wik: "Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. "
The point is that those finite decompositions require an uncountable number of choices, so you first need to have an infinite amount of sets to chose the points from. The decompositions are still finite though.
@@Robertgancadiameter and density don't change. (I don't think something like density exists in this context, its either solid or its not. In this case the starting ball and the ending balls are all solid)
@@GuzMat-matematicas a single point has measure 0, you won't increase the measure of the ball by rotating and reassembling sets of measure 0. You need non measurable sets for this which the axiom of choice implies the existence of.
@@Robertganca That's a good question, because this paradox questions our understanding of volume and density. The problem here is that the axiom of choice allows you to contruct sets, where assigning any value as "volume" to it would result in a contradiction. In measure theory we call these sets non-measurable sets and by allowing the axiom of choice, we have to accept them. The trick in Banach Tarski's paradox is to split the ball into unmeasurable sets in order to circumvent any volume restrictions.
TED-Ed math videos are always impeccable. There's a great ending quote on Spivak's Calculus from Jonathan Swift, when he lays the definitions of the reals: There was a most ingenious Architect who had contrived a new Method for building Houses, by beginning at the Roof, and working downwards to the Foundation.
Let's not forget during the Banach-Tarski construction, the pieces the ball is cut into are in fact non-measurable, meaning there's no consistent way to assign a volume to each of them, making it even less realistic
I think it's better to go read some academic paper that explains what Banach-Tarski really is than watching this video that tried but failed to simplify this whole thing.
the most beautiful thing about math to me is how well we've learned to talk about imaginary things with other people - this video boils this down and does it in an accessible and fun way too :) bravo to the makers!
I only heard the name of this Banach-Tarski paradox and didn't really know what it was, but the animation definitely helped me understand what it was! 👍
I was following along, then I got lost partway through. It’s a good thing I understand the math needed for everyday life! Not that I was terrible at math in school, it’s just not something I need day to day.
One of my biggest math questions is about the order of operations. In my mother's time in school, all math was answered in the order it was written. For example; 1+2×3 would be 9. But when I was in school, we worked off BEDMAS. Using BEDMAS, the same question; 1+2×3 would end up with the answer being 7. How could the fundamental nature of math change in half a century and not throw the world into chaos?
No, I think, the problem with the order of operations is caused by our teachers not being careful and fully informed. Besides, to make sure there is no ambiguity, we can use parenthesis and never skip a "x" when we imply multiplication. Take up a programming language and you will see how easy and clear the operations are.
My interpretation is that underlying fundamental maths hasn’t changed, only the notation that’s used. As long as you have agreed the notation (e.g. order of operations, symbols, or base numbers) with the person you are communicating with, no problem is caused.
It makes me wonder if the axioms we choose can get closer to supporting all math, or is it actually just building different interesting homes. is one set of axioms "better" than the other or just simply different? I think it would be a great idea if we link back to ted's video on godel and "is math created or discovered?"
That's a great question! Unfortunately there's no axioms you could choose which would support "all" of mathematics. For example, whatever collection of axioms you choose, there are always theorems you can't prove using those axioms--you can never "have enough", so to speak. Some collections of axioms are "stronger" than others, in the sense that everything you could build on one you could build on the other, but none are better, just different :)
@@hiredfiredtired I defer to you, as I am not a math person. Still Euclid’s axiom was found not to be needed in certain situations. Perhaps AoC is similar? But I think I see what you’re saying. AoC is perfect and just because a paradoxical result is unintuitive doesn’t mean it’s wrong?
@@Otis151 My point is that you are stuck between having banac taramy and well ordering, and NOT having the fact that you can pick an item from a collection of nonempty sets. The finite axiom of choice is actually provable without axiom of choice, axiom of choice is really just for infinities. Thats why theres so many weird results from it
@@Otis151I'm just an interested layman here but my understanding is that, without the axiom of choice, there are important (to us) mathematical questions that don't have an answer; so we can either just give up on those questions, or accept the sometimes bizarre answers that using the axiom gives us. there are weaker axioms that can be chosen in place of the axiom of choice, but as far as I'm aware no one's found a way to get the answers without the weirdness. the vast majority of mathematicians just accept the axiom of choice and its consequences and get on with their work
Great question! In a way the axiom of choice is very strong: it can be used to prove a lot of theorems, some weird, some non-weird. Your question essentially is, "can the axiom of choice be weakened to produce fewer weird results while preserving the non-weird ones?" Unfortunately the answer is, essentially, no; or at least, we haven't found such a weakening. The most popular weakening is called the "Axiom of Dependent Choice" (DC). Taking DC instead of AC avoids a lot of weirdness: Banach-Tarski cannot be proved using DC (more generally, DC cannot be used to show the existence of non-measurable sets). And lots of good math can be done with just DC (most of real analysis, so most of calculus, for examples). But some really important theorems cannot be proved using just DC. In functional analysis, the Hahn-Banach and many other theorems require AC and not DC, and in measure theory many fundamental theorems (even the sigma-additivity of the Lebesgue measure!) cannot be shown using DC. So you avoid lots of weirdness but you also loose lots of important theorems.
Great video, but your explanation for the axiom of choice (2:20) was rather unclear - I’m already familiar with the aoc and still got lost in the metaphor. It might have been better to explain what the axiom actually is, before saying when a choice is valid and telling the story about the omniscient chooser.
Not really, the main point of this paradox is to show that there is no consistent way to assign a measure (i.e. volume) to every possible subset of three dimensional space, because otherwise you can transform a ball into two balls of the same size through seemingly volume-preserving transformations. Hilberts paradox is not really a paradox as it simply shows a few (unintuitive) differences between finite and infinite sets
Because math is a man-made tool rather than a natural science, it can contain such examples that likely have no parallel in the physical world. To my understanding this paradox stems down to being able to break down a set of values (i.e. ones that represent a sphere) into an infinite series of an infinitely-high resolution. The unobservable universe may very well be infinite, but applied physical situations within our observable reality don't seem to be infinite. If a physical sphere is made of a finite number of subatomic particles, and space as well may be of a finite resolution, we can't section a physical one an infinite number of times to make use of the mathematical phenomenon which is ∞=∞-1=∞-2... Point being, this paradox breaks our brain because it applies to a mathematical sphere values that don't exist in a physical sphere.
Consider a hypothetical scenario, that in our universe we get a new law that if we take 2 and more 2 object and then add them then they will collapse and turn into 0. Now in this case will you say 2+2=0 just because we are seeing in the universe when 2 and 2 objects get add they collapse and turn into 0? Or you will say 2+2=4 because of logical consistency?
We wouldn't change the way + behaves, but we would make a new operation that is consistent with the new behavior of objects in our universe. So maybe 2#2=0
Zeno's paradox tells us that if we take steps towards a door way such that each consecutive step is half that of the previous step, them we will never reach our destination. In reality we do due to convergence. In reality measurement is limited as shown by Heisenberg's uncertainty principle and paradoxes arise when we try to use "common" logic with infinities. They do not work because infinity is a concept, not a number and is therefore immeasurable AND it does not fulfil the axioms of real numbers.
It appears that you lack knowledge of what you are trying to talk about. An example of a measure that is talked about here is the lebesgue-measure. It is indeed possible to measure infinite sets, just not all of them (unless you use the zero-measure) if you assume the axiom of choice.
@@julianbruns7459 You are absolutely right, I'm not a mathematician. However, semantics of "infinity" and "immeasurable" aside, I was referring to the application of the axioms of real numbers to infinity. I didn't mention lebesgue-measures, nor infinite sets, nor did I invoke the axiom of choice. Nevertheless, I stand by your greater wisdom.
@@samshort365 oh okay. I was merely assuming that because i thought your comment was related to the video/the banach tarski paradox. If your message was that you can't treat infinity as a real number and have to be careful when talking about it, i completely agree. Our intuition often fails when carelessly talking about infinity.
Wouldn’t the Heisenberg Uncertainty Principle make the Axiom of Choice illogical for analyzing physical geometry? Particularly when dealing with infinitely small pieces.
The fact that we can't know precisely the position and momentum of elementary particles is not relevant here i think. (Also i don't think quantum mechanics assumes infinitely small pieces). Most people would agree that the number of elementary particles in a given sphere is finite (or at most countably infinite). Even if you interpret those particles as sets, the banach tarski paradox doesn't apply, because the axiom of choice needs an uncountably infinite amount of points to create this effect.
Math is supposed to be abstract. One thing that strikes people about it is that it can never be completely understood. Even when you look at it in a different angle, there's still some areas that need analysis on. Equations are anything but perfect. People spend years just looking for the 'correct' answers when they probably aren't the best answers.
Your language is confusing. What do you mean by "size"? Do you mean cardinality? In that case it is false, since both the integers and natural numbers have the same cardinality. Do you mean lebesgue measure? Then this is trivially not true. Are you referring to the generalized continuum hypothesis? Then the Axiom of Choice isn't necessary for proving it, in fact it is the exact opposite: ZF+ GCH implies AoC. Do you mean Cantors theorem, that the power set of a set has larger cardinality than that set? Then your communication would have been pretty poorly. Could you please elaborate what you mean?
So by following the BT-paradox, you could slice the people who onderstand the BT-paradox into infinitely many pieces and construct twice that many people.
I learned basic math with apples and oranges, and of course I understand irrational numbers, but if the elements and variants have to be ordeal and not existing, then how can I know what you say is correct?
When I saw the name Bancah-Tarski i immediately thought of the riddle ted ed did.The infinite gold riddle Where the name was in front of the little mans shirt.
Until we discover how our universe can support... - Infinitely sharp knives, - Infinitely divisible balls, and - Processes for being able to complete infinite numbers of actions in a finite amount of time... surely this is all moot? If we can accept the existence of, say, the square root of -1 (which we can manipulate mathematically but not manifest physically) then why should this be any different?
The B-T paradox is not so controversial. The set of points in the unit sphere is unaccountably infinite - effectively the same number (cardinality) of points as two times the unit sphere. So no reason why you shouldn't be able to create two from one. After all it's not hard to create a function mapping the integers to the seemingly much larger set that is all the rational numbers. Why is this any different?
140K views, and only 279 comments? That's stunning Yeah, so I'm a mathematician And I already have a complaint At 1:27, the soulless voice says _that adding zero to a number has no effect is an axiom_ Which it denotes by *_x + 0 = x_* I suppose it's possible to assert that this is an axiom But a much better formulation is that this is the _definition of _*_zero_*_ in an algebra system_ Where the binary operation in this algebraic system is *addition* commonly represented by the symbol *+*
Sometimes you can follow all the rules and come to an unexpected result. This is like California declaring the bee a fish because it meets all the criteria. No it's not actually a fish but according to the rules we established it is legally a fish. The same goes for this. It's not technically possible in reality (as far as we know), however according to the rules we have established it's possible.
Whilst I appreciate the message at the end about axioms being potentially non-universal, please could we not give our infinitely-sharp knife to an alien
Not sure if right, but I think I came up with a version of Banach-Tarski Paradox of my own. Take a series of whole numbers from 1 to infinity. Now take all the odd numbers in the series and add it up. You'll find that the total is infinity. Now do the same with even numbers. Add all the even numbers up and you'll get infinity too. In the end you get two infinities from one infinity. Which doesn't seem right considering the odd version of infinity doesn't have all the whole numbers yet it still totals to infinity.
The size of a set in the sense you are talking about is defined using the concept of cardinality: if there is a bijection between two sets , they are said to have equal cardinality. The set of even numbers has the same cardinality as the set of natural numbers. Likewise, there even exists a bijection between the sets of prime numbers and the set of rational numbers. The set of real numbers is larger in the sense that there is no bijection between it and the naturals, the reals are also called an uncountable set. The kind of measure the video is talking about is different. It assigns an infinite set of points a finite value. For example the interval [0,1] wich has an uncountable number of points gets assigned the value 1. The point of the banach-tarski paradox is that some sets can't get a value assigned in this sense and thus rotating and reassembling a ball into those sets enables you to "double the ball" in a specific way.
This could just explain why unification of quantum theory and standard model is stuck. Historically, about every time something has been kept for its practicality over accuracy has been proven a terrible mistake
No, i don't think you can simplify it like that without making it plain wrong. If you really want to put it into one sentence, maybe say something like: "the axiom of choice implies the existence of non-measurable sets".
Your axiom system needs to be consitent, otherwise every theorem can be proven by the principle of explosion, and thus also every "true" statement. Your axioms also need to be recursively enumerable, for example i think "true arithmetic" trivially proves all true statements in arithmetic but is not recursively enumerable. The axiom system also needs to be able to interpret a certain amount of arithmetic for the proof to work, for example this doesn't apply to presburger arithmetic wich is decidable. Lastly, it is hard to define what make a sentence "true", so it is better to say that a formal system with the said requirements isn't negation complete, i.e. there are sentences A where it can neither proof A nor the negation of A. Usually we would want at least one of them to have the name "true", which is where your formulation comes from. We don't construct a sentence that we show is true without proving it though.
Math without the axiom of choice? No thanks, it's way too important. The equivalent Zorn's lemma is the indispensable tool in the Swiss army knife of math. Take the Hahn-Banach theorem as an example, there would almost be no functional analysis whithout it.
I watched the Vsauce video on the Banach-Tarski Paradox about 4 times before somewhat grasping the concept...
Thats the one Vsauce video I simply cannot understand
I watched it when I was binge watching all his videos for the first time when I was 11. I remember being genuinely dizzy after that.
He went into a lot of the technical detail, but the most important base principle is
♾️ x 2 = ♾️
The practical-sounding description of “cutting up a ball and putting the pieces together into two balls” disguises the fact that at its core, this is like Hilbert’s Infinite Hotel
Us
OR DID YOU?!
I first encountered the Banach-Tarski paradox in my subject for mathematical proof. When we discussed certain set-theoretical concepts, we naturally covered the Axiom of Choice. Our teacher introduced us to the Banach-Tarski paradox and promised we would eventually learn its proof as we attended higher mathematical classes. I needed to learn concepts from mathematical analysis and topology to actually understand the proof.
What do I have to study to even just understand what this problem is about?
@@Schmidtelpunkt If you read the proofs of the Banach-Tarski paradox, you'll need extensive knowledge on group theory, set theory, analysis, and linear algebra. The Axiom of Choice is usually introduced in foundational math courses (usually where principles of mathematical proof is introduced).
@@Schmidtelpunkt This proof is longer, but is more detailed. Plus, I love how the author manages to explain some of the concepts.
www.diva-portal.org/smash/get/diva2:1672461/FULLTEXT01.pdf
@@Schmidtelpunkt The Axiom of Choice has such incredibly diverse and apparently formulations
I once wrote a paper which merely listed these various formulation, with a discussion of how bizarrely unconnected these are
I'll take a stab at it by calling it the mathematician's get out of jail free card for proof-writing - with the realization that it's an adequate conceptualization
@@JaybeePenaflor No topology? I assumed BTP would require it. It _sounds_ like topology to me
Does maths have a fatal flaw?
Yes, it makes my head hurt
My brain hurts when it comes to math because I have Dyscalculia and math is a foreign language that I can't ever understand.
@@92RKIDwth even is that?
Maths make my headache go away I do it as a hobby sometimes
@@savitatawade2403
its like Dyslexia, but specific to numbers
whose flaw is that
my head hurts just thinking about the video 😭😭😭 but the animation is adorable omg
Me too 😂
In fact, the Banach-Tarski paradox is an abbreviation. Full name is the Banach-Tarski Banach-Tarski paradox paradox
I see what you did there 🗿
Or the BTBTPP?
Wouldn't that make it the Banach-Tarski^n paradox^n, for arbitrarily high values of n, via iteration of the process?
So it's a pair o' paradoxes?
You haven't explained why the axiom of choice makes the sphere construction possible
It makes it possible to deconstruct the (measurable) ball into non measurable sets that, if reassembled into a (measurable) set, happen to have a different volume. I don't think explaining that would fit into a Ted video aimed at a general audience.
if you have no mathematical knowledge besides highschool, it would probably take you around a year of studies that are necessary to understand the proof
Yes, because the tiny fraction of the audience that would understand it would be all made of people who already understood it.
I guess one of the goals here is to inspire people, in particular young people, to seek that kind of knowledge. But the technical parts of it require years of intense study, of course. Anyway, understand the basics of mathematics, as in what is an axiom and what is a theorem, is much more accessible and was indeed covered here.
@@julianbruns7459 By measurable, you mean in the Lebesgue integration sense?
Dear lord, that was my waterloo
This from a person whose area of expertise is the pinnacle of an introductory graduate level course in abstract algebra
Namely Galois Theory
@@נועםדוד-י8ד I'll postulate that it would take anyone - under those condition - a heck of a lot longer
Unless they were extremely gifted in math
In which case, they'd have known that already and therefore have much more than a basic understanding
I’ve always loved that futurama references this with the professor’s duplicator machine in the episode where there are infinite benders.
Giving an alien you've just met an infinitely sharp knife might not be the smartest idea
I was thinking the same ..
Giving A.I the tools and compute required to run the whole world's economy is undoubtedly a good idea according to industry leaders
Nope. I don't understand. Have a nice day
Yeah, this was a very poorly designed Ted Ed video. Usually more practical examples in other Ted Ed videos.
@@stephenj9470 Or maybe we were not on a level of understanding 👀
This is to advanced
The Banach Tarski paradox is that its theoretically possible for there to be an infinite number of options and the same option to be picked every time. If you have a bag of M&Ms and put each M&M into a different box in a room with infinity number of boxes its theoretically possible for an all knowing being to choose those same M&Ms each time.
@@sackeshiIs an all knowing being possible though?
What's an anagram of Banach-Tarksi?
Banach-Tarski Banach-Tarski
yes.
I studied math in college and you guys explained the axiom of choice so clearly that I learned something new
I loved this video! Animations are always on point. Learning about axioms in college was so complicated, but you guys made it so easy to digest.
the animation is ON POINT. beautifully done
Wait, I actually was able to follow through with this! Basically, math itself is pretty abstract but it becomes concrete once we apply it in a practical situation (a.k.a. reality). And there are alot of alternate "truths", I guess, that would lead to different realities. Cool stuff.
One of my favourite videos to date, loved the animation as well as the analogies used!
Ted ed finally doing a video on this, nice🔥🔥
Not an infinite number (at 0:15), a finite number. From Wik: "Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. "
If it’s finite, then how are you able to recreate the ball twice with the same diameter and density? Or do those change?
The point is that those finite decompositions require an uncountable number of choices, so you first need to have an infinite amount of sets to chose the points from. The decompositions are still finite though.
@@Robertgancadiameter and density don't change. (I don't think something like density exists in this context, its either solid or its not. In this case the starting ball and the ending balls are all solid)
@@GuzMat-matematicas a single point has measure 0, you won't increase the measure of the ball by rotating and reassembling sets of measure 0. You need non measurable sets for this which the axiom of choice implies the existence of.
@@Robertganca
That's a good question, because this paradox questions our understanding of volume and density.
The problem here is that the axiom of choice allows you to contruct sets, where assigning any value as "volume" to it would result in a contradiction.
In measure theory we call these sets non-measurable sets and by allowing the axiom of choice, we have to accept them.
The trick in Banach Tarski's paradox is to split the ball into unmeasurable sets in order to circumvent any volume restrictions.
TED-Ed math videos are always impeccable. There's a great ending quote on Spivak's Calculus from Jonathan Swift, when he lays the definitions of the reals:
There was a most ingenious Architect
who had contrived a new Method
for building Houses,
by beginning at the Roof, and working
downwards to the Foundation.
Michael had already done a great job in explaining this Paradox.
This is like a trailer for that 22min abomination
Well it turns out this video isn't really about the paradox, but makes a bigger point about its implications for the bases of mathematics
Let's not forget during the Banach-Tarski construction, the pieces the ball is cut into are in fact non-measurable, meaning there's no consistent way to assign a volume to each of them, making it even less realistic
I think that's kinda key, right? It's missing an axiom we use to model reality as we perveive it, right?
Yea that's the real problem. If you use pointless topology than such pathological result will not exist
I think it's better to go read some academic paper that explains what Banach-Tarski really is than watching this video that tried but failed to simplify this whole thing.
What can you say about the vsauce video about this paradox?
As a lover of mathematics this video is really amazing great job
Can't find anyone talking the animation here, I think it makes the video much easier to comprehend!
the most beautiful thing about math to me is how well we've learned to talk about imaginary things with other people - this video boils this down and does it in an accessible and fun way too :) bravo to the makers!
Couldnt understand
I only heard the name of this Banach-Tarski paradox and didn't really know what it was, but the animation definitely helped me understand what it was! 👍
We learned this in my really analysis class about a month ago!
Quite an intriguing concept!
This reminds me of Non-Euclidean video games like Antichamber and Superliminal.
The graphics of this episode are excellent, particularly the math houses with differing foundations.
Damn!
YOU ARE PERFECT in making simple paradox more complicated 😒
If you find this explanation complicated, how can you find the actual paradox simple?
I wanted a video on this topic for a long time. You guys reading my minds! 😭 Maths hasn't reached a perfect stage *yet* .
I was following along, then I got lost partway through. It’s a good thing I understand the math needed for everyday life! Not that I was terrible at math in school, it’s just not something I need day to day.
Flaws? Yes, it's not always fun to learn it.
The catch with this and with Gödel is that a copy (or an infinite number of copies) is being made from the infinite
One of my biggest math questions is about the order of operations. In my mother's time in school, all math was answered in the order it was written.
For example; 1+2×3 would be 9.
But when I was in school, we worked off BEDMAS. Using BEDMAS, the same question; 1+2×3 would end up with the answer being 7.
How could the fundamental nature of math change in half a century and not throw the world into chaos?
No, I think, the problem with the order of operations is caused by our teachers not being careful and fully informed. Besides, to make sure there is no ambiguity, we can use parenthesis and never skip a "x" when we imply multiplication. Take up a programming language and you will see how easy and clear the operations are.
My interpretation is that underlying fundamental maths hasn’t changed, only the notation that’s used. As long as you have agreed the notation (e.g. order of operations, symbols, or base numbers) with the person you are communicating with, no problem is caused.
The Axiom of Choice(AC) can be substituted by the Axiom of Determinacy(AD).
Explain
It makes me wonder if the axioms we choose can get closer to supporting all math, or is it actually just building different interesting homes. is one set of axioms "better" than the other or just simply different? I think it would be a great idea if we link back to ted's video on godel and "is math created or discovered?"
That's a great question! Unfortunately there's no axioms you could choose which would support "all" of mathematics. For example, whatever collection of axioms you choose, there are always theorems you can't prove using those axioms--you can never "have enough", so to speak. Some collections of axioms are "stronger" than others, in the sense that everything you could build on one you could build on the other, but none are better, just different :)
Axiom of Choice sounds like a powerful spirit!
Beautifully done
Could we refine the AoC so that it’s usable in the sensical applications while not applicable in known non-sensical?
theres no such thing as "sensical applications". Axiom of choice naturally leads to these things.
@@hiredfiredtired I defer to you, as I am not a math person.
Still Euclid’s axiom was found not to be needed in certain situations. Perhaps AoC is similar?
But I think I see what you’re saying. AoC is perfect and just because a paradoxical result is unintuitive doesn’t mean it’s wrong?
@@Otis151 My point is that you are stuck between having banac taramy and well ordering, and NOT having the fact that you can pick an item from a collection of nonempty sets. The finite axiom of choice is actually provable without axiom of choice, axiom of choice is really just for infinities. Thats why theres so many weird results from it
@@Otis151I'm just an interested layman here but my understanding is that, without the axiom of choice, there are important (to us) mathematical questions that don't have an answer; so we can either just give up on those questions, or accept the sometimes bizarre answers that using the axiom gives us.
there are weaker axioms that can be chosen in place of the axiom of choice, but as far as I'm aware no one's found a way to get the answers without the weirdness.
the vast majority of mathematicians just accept the axiom of choice and its consequences and get on with their work
Great question! In a way the axiom of choice is very strong: it can be used to prove a lot of theorems, some weird, some non-weird. Your question essentially is, "can the axiom of choice be weakened to produce fewer weird results while preserving the non-weird ones?" Unfortunately the answer is, essentially, no; or at least, we haven't found such a weakening.
The most popular weakening is called the "Axiom of Dependent Choice" (DC). Taking DC instead of AC avoids a lot of weirdness: Banach-Tarski cannot be proved using DC (more generally, DC cannot be used to show the existence of non-measurable sets). And lots of good math can be done with just DC (most of real analysis, so most of calculus, for examples). But some really important theorems cannot be proved using just DC. In functional analysis, the Hahn-Banach and many other theorems require AC and not DC, and in measure theory many fundamental theorems (even the sigma-additivity of the Lebesgue measure!) cannot be shown using DC. So you avoid lots of weirdness but you also loose lots of important theorems.
excellent explanation and presentation!
Great video, thanks
Great video, but your explanation for the axiom of choice (2:20) was rather unclear - I’m already familiar with the aoc and still got lost in the metaphor.
It might have been better to explain what the axiom actually is, before saying when a choice is valid and telling the story about the omniscient chooser.
A "cameo" from Heptapods "Flapper" or "Raspberry" at the end would've been an amazingly apt reference.
The flaw is that I can’t do math
It's a similar concept to Hilbert paradox
Not really, the main point of this paradox is to show that there is no consistent way to assign a measure (i.e. volume) to every possible subset of three dimensional space, because otherwise you can transform a ball into two balls of the same size through seemingly volume-preserving transformations. Hilberts paradox is not really a paradox as it simply shows a few (unintuitive) differences between finite and infinite sets
@@dogedev1337Thanks you explained it beautifully
Because math is a man-made tool rather than a natural science, it can contain such examples that likely have no parallel in the physical world.
To my understanding this paradox stems down to being able to break down a set of values (i.e. ones that represent a sphere) into an infinite series of an infinitely-high resolution.
The unobservable universe may very well be infinite, but applied physical situations within our observable reality don't seem to be infinite. If a physical sphere is made of a finite number of subatomic particles, and space as well may be of a finite resolution, we can't section a physical one an infinite number of times to make use of the mathematical phenomenon which is ∞=∞-1=∞-2...
Point being, this paradox breaks our brain because it applies to a mathematical sphere values that don't exist in a physical sphere.
How about a video on the strength/weakenss of logic amd rules of inference.
You had me at 'Maths' Ted-ed. Love your videoes on maths.
Great video, didn’t understand any of it.
Hey Ted -ed sugestion make a video about the 1992 riots of Los Angeles.
That goes against axiom of greatness of multiculturalism
@@blazer9547the one axiom that still remains even where there is proof that says otherwise 😂
Fascinating! Thank you!
I watched two different videos several times, I'll finally get it
its usually is a rounding error
I'm not smart enough to understand this but give the animation team a raise they did an amazing job
Consider a hypothetical scenario, that in our universe we get a new law that if we take 2 and more 2 object and then add them then they will collapse and turn into 0. Now in this case will you say 2+2=0 just because we are seeing in the universe when 2 and 2 objects get add they collapse and turn into 0? Or you will say 2+2=4 because of logical consistency?
We wouldn't change the way + behaves, but we would make a new operation that is consistent with the new behavior of objects in our universe. So maybe 2#2=0
@@לוטם-ו1ע mod 4 arithmetic still uses a + sign.
Good video.
Zeno's paradox tells us that if we take steps towards a door way such that each consecutive step is half that of the previous step, them we will never reach our destination. In reality we do due to convergence. In reality measurement is limited as shown by Heisenberg's uncertainty principle and paradoxes arise when we try to use "common" logic with infinities. They do not work because infinity is a concept, not a number and is therefore immeasurable AND it does not fulfil the axioms of real numbers.
It appears that you lack knowledge of what you are trying to talk about. An example of a measure that is talked about here is the lebesgue-measure. It is indeed possible to measure infinite sets, just not all of them (unless you use the zero-measure) if you assume the axiom of choice.
@@julianbruns7459 You are absolutely right, I'm not a mathematician. However, semantics of "infinity" and "immeasurable" aside, I was referring to the application of the axioms of real numbers to infinity. I didn't mention lebesgue-measures, nor infinite sets, nor did I invoke the axiom of choice. Nevertheless, I stand by your greater wisdom.
@@samshort365 oh okay. I was merely assuming that because i thought your comment was related to the video/the banach tarski paradox. If your message was that you can't treat infinity as a real number and have to be careful when talking about it, i completely agree. Our intuition often fails when carelessly talking about infinity.
I am in season 3 of this video and it's tough but I think I'll eventually understand.
So different axioms lead to different results. How do we define these axioms?
Yea I most DEFINITELY understood that.
Thanks Ted. This was actually quite beautiful
This flew over my head
The Banach Tarski paradox, will complete me.
🔄❤
Babe wake up..Ted-Ed just dropped another banger
Wouldn’t the Heisenberg Uncertainty Principle make the Axiom of Choice illogical for analyzing physical geometry? Particularly when dealing with infinitely small pieces.
The fact that we can't know precisely the position and momentum of elementary particles is not relevant here i think. (Also i don't think quantum mechanics assumes infinitely small pieces).
Most people would agree that the number of elementary particles in a given sphere is finite (or at most countably infinite). Even if you interpret those particles as sets, the banach tarski paradox doesn't apply, because the axiom of choice needs an uncountably infinite amount of points to create this effect.
Heisenberg is physics, BT is maths. You’re in the clear :)
I love this so much! My brain feels expanded
Love these vids, keep it up :)
Math is supposed to be abstract. One thing that strikes people about it is that it can never be completely understood. Even when you look at it in a different angle, there's still some areas that need analysis on. Equations are anything but perfect. People spend years just looking for the 'correct' answers when they probably aren't the best answers.
AoC is necessary for proving that if two sets aren't the same size, one of them is bigger
Your language is confusing. What do you mean by "size"? Do you mean cardinality? In that case it is false, since both the integers and natural numbers have the same cardinality. Do you mean lebesgue measure? Then this is trivially not true.
Are you referring to the generalized continuum hypothesis? Then the Axiom of Choice isn't necessary for proving it, in fact it is the exact opposite: ZF+ GCH implies AoC. Do you mean Cantors theorem, that the power set of a set has larger cardinality than that set? Then your communication would have been pretty poorly.
Could you please elaborate what you mean?
@@julianbruns7459 i mean cardinality
in both the "bigger" part and the "size" part.
@@MichaelDarrow-tr1mn then this seems more like a tautology that you don't need the axiom of choice for, just the definition of an order.
@@julianbruns7459 pretty sure it's actually equivalent to the axiom of choice
@@MichaelDarrow-tr1mn would you be so kind and give me a source for that claim?
Very nice video
So by following the BT-paradox, you could slice the people who onderstand the BT-paradox into infinitely many pieces and construct twice that many people.
I will not be handing an alien a sharp knife.
I learned basic math with apples and oranges, and of course I understand irrational numbers, but if the elements and variants have to be ordeal and not existing, then how can I know what you say is correct?
When I saw the name Bancah-Tarski i immediately thought of the riddle ted ed did.The infinite gold riddle Where the name was in front of the little mans shirt.
Until we discover how our universe can support...
- Infinitely sharp knives,
- Infinitely divisible balls, and
- Processes for being able to complete infinite numbers of actions in a finite amount of time...
surely this is all moot?
If we can accept the existence of, say, the square root of -1 (which we can manipulate mathematically but not manifest physically) then why should this be any different?
Banach-Tarski? Like the Banach-Tarski from the Infinite Gold riddle?
That's what came to my mind as well.
@@thenovicenovelist Really?
The Banach-Tarski paradox in set theory.
The B-T paradox is not so controversial. The set of points in the unit sphere is unaccountably infinite - effectively the same number (cardinality) of points as two times the unit sphere. So no reason why you shouldn't be able to create two from one.
After all it's not hard to create a function mapping the integers to the seemingly much larger set that is all the rational numbers. Why is this any different?
140K views, and only 279 comments?
That's stunning
Yeah, so I'm a mathematician
And I already have a complaint
At 1:27, the soulless voice says
_that adding zero to a number has no effect is an axiom_
Which it denotes by
*_x + 0 = x_*
I suppose it's possible to assert that this is an axiom
But a much better formulation is that this is the _definition of _*_zero_*_ in an algebra system_
Where the binary operation in this algebraic system is *addition* commonly represented by the symbol *+*
The only thing I could get was that Crocker was right 2+2 might be equal to 🐟 after all
0:02 - And then there is me who doesn't know what an analogy is... And I was the best at math in my school...
This is exactly why I LOVE SCIENCE 😭
Sometimes you can follow all the rules and come to an unexpected result. This is like California declaring the bee a fish because it meets all the criteria. No it's not actually a fish but according to the rules we established it is legally a fish.
The same goes for this. It's not technically possible in reality (as far as we know), however according to the rules we have established it's possible.
Well in our universe it is not possible. In another one it may be. And no, this isn't sci-fi. 😂
Did you get any of that?
We stan the axiom of choice
Whilst I appreciate the message at the end about axioms being potentially non-universal, please could we not give our infinitely-sharp knife to an alien
Not sure if right, but I think I came up with a version of Banach-Tarski Paradox of my own.
Take a series of whole numbers from 1 to infinity. Now take all the odd numbers in the series and add it up. You'll find that the total is infinity. Now do the same with even numbers. Add all the even numbers up and you'll get infinity too.
In the end you get two infinities from one infinity. Which doesn't seem right considering the odd version of infinity doesn't have all the whole numbers yet it still totals to infinity.
The size of a set in the sense you are talking about is defined using the concept of cardinality: if there is a bijection between two sets , they are said to have equal cardinality. The set of even numbers has the same cardinality as the set of natural numbers. Likewise, there even exists a bijection between the sets of prime numbers and the set of rational numbers. The set of real numbers is larger in the sense that there is no bijection between it and the naturals, the reals are also called an uncountable set.
The kind of measure the video is talking about is different. It assigns an infinite set of points a finite value. For example the interval [0,1] wich has an uncountable number of points gets assigned the value 1.
The point of the banach-tarski paradox is that some sets can't get a value assigned in this sense and thus rotating and reassembling a ball into those sets enables you to "double the ball" in a specific way.
This could just explain why unification of quantum theory and standard model is stuck. Historically, about every time something has been kept for its practicality over accuracy has been proven a terrible mistake
So that’s where the little man from the Infinite Coin Riddle got his name. Very interesting.
The mysterious omniscient chooser, as a mathematician, I can't stop laughing; this analogy was too good.
The animations are so good that they are distracting from the actual narration. had to watch twice.
So basically the paradox in simple terms is ((1/infinite)x(infinite))=2?
No, i don't think you can simplify it like that without making it plain wrong. If you really want to put it into one sentence, maybe say something like: "the axiom of choice implies the existence of non-measurable sets".
No matter what axioms you choose, there will always be true statements that you will be unable to prove.
Almost! Its true if your axioms are strong enough for a portion of standard arithmetic. For weaker axiom systems it can be different.
@@stefanperko That sounds familiar, but makes no sense to me. I’ve been out of school for too long lol
Your axiom system needs to be consitent, otherwise every theorem can be proven by the principle of explosion, and thus also every "true" statement. Your axioms also need to be recursively enumerable, for example i think "true arithmetic" trivially proves all true statements in arithmetic but is not recursively enumerable. The axiom system also needs to be able to interpret a certain amount of arithmetic for the proof to work, for example this doesn't apply to presburger arithmetic wich is decidable.
Lastly, it is hard to define what make a sentence "true", so it is better to say that a formal system with the said requirements isn't negation complete, i.e. there are sentences A where it can neither proof A nor the negation of A. Usually we would want at least one of them to have the name "true", which is where your formulation comes from. We don't construct a sentence that we show is true without proving it though.
me before watching the video - what is Banach-Tarski paradox ?🤥
me after watching the video - what is Banach-Tarski paradox ?🤥
Math without the axiom of choice? No thanks, it's way too important. The equivalent Zorn's lemma is the indispensable tool in the Swiss army knife of math. Take the Hahn-Banach theorem as an example, there would almost be no functional analysis whithout it.
So, at last, in this video what the heck is axiom of choice? Why they just mentioned but didn't explain about it?
They did explain it, using the "marbles in boxes" metaphor.
Ahhhh high school math class flashbacks. Didn't understand much and I zoned out.
I swear mathematicians just enjoy making confusing problems that dont really make sense 😭
Oh... NOW I understand the Strangle Little Man with the bag that makes infinite coins