Cantor's Infinity Paradox | Set Theory

[@upandatom]

I see there is a lot of confusion over the first proof of the uncountability of the Real numbers. This is my fault. I was unclear and didn't provide the whole proof. I have made a video explaining it from start to finish in detail. Sorry for the confusion. kzbin.info/www/bejne/laLcdXWLpaZ6q9E

[@getowtofheyah3161]

Up and Atom Mathemtics PhD student who has taken axiomatic set theory. Lots that needs to be corrected in your video.

[@monad_tcp]

oh, now it makes sense

[@pat6595]

Jade, do you think there will be a Theory of Everything?

[@daredevil016]

Hey Jade 👋 , Since we know that there are infinite no. between any two real no. then what is the no. just next to zero(0) ?Or a no. i.e. just bigger than zero(0)?? If there are infinite no.ahead of it ,what no. it would be ??? Btw I m high school student from India , if there is any mistake in my question that I have mentioned ,please lemme know & please give your valuable opinion on this ...anyways new subscriber love your videos ,your way of explaining it ,keep making videos this help me alot in learning ...sorry for too long comment 😅.

[@AgentOccam]

@@daredevil016 there's no number "just next to" zero, in real number terms. Assume the phrase "just next to" refers to the only number, call it x, larger than zero, of which it is true that you can't define a number that's less than x but still greater than zero. Can you do this with the natural numbers? Yes. For example, 3 has exactly one number that can be x, namely 4. No number that meets the definition of a natural number can be less than 4 and yet greater than 3. So in those terms, 4 is "just next to" 3. So in that sense, 1 is the number "just next to" 0. But there's no number in the real numbers, say, that can meet that definition.

[@palrob1714]

I love how goofy your drawings are. They look very endearing.

[@therayven3147]

@Joe Tonner amen to that...

[@aurelienyonrac]

I love here drawings. It takes a pure soul to draw like that. I hope she uses them in all of here video 😍

[@dubber889]

mathematician comes to a bar, asking for a 1 litre of beer and then asking half of it, and half of it, and then half of it. The bartenders then said, "You should know your limit".

[@savagenovelist2983]

ryuzaki Can you not even tell the joke correctly? An infinite number of mathematicians walk into a bar: The first one asks for a shot, the second one for half a shot, the third for a third of a shot... At some point, the bartender hands them two shots and says: “Know your limits.”

[@bigmichael6156]

@@savagenovelist2983 what if 64 mathematicians go to a chinese restaurant. Asked what they whant to eat. First one said: 1 grain of rice, please. the second : two grain third whants four next whants eight and so on? Chef : we must collect rice harvest for ten years to serve the last.

[@Spacetauren]

@@savagenovelist2983 Actually the sum (for n from 1 -> infinity) of 1/n diverges to infinity. It would be more like : An infinite number of mathematicians walk into a bar: The first one asks for a shot, the second one for half a shot, the third for a fourth of a shot, the fourth for an eight of a shot... At some point, the bartender hands them two shots and says: “Know your limits.”

[@desmondroberts6034]

@@savagenovelist2983 Why wouldn't the bartender say in impatience: 'Here - knock yourselves out'

[@HarshPatel27]

@@bigmichael6156 That's a version of the famous Indian tale of 'rice on a chess board' kzbin.info/www/bejne/bWLcpntpa5qFq8k

[@legoharry100]

"My infinity is bigger than your infinity." - Thanos

[@jeromedavies2408]

Indeed, is the Infinity Gauntlet enumerable or non-enumerable?

[@lukasvozak7698]

Don't wanna destroy 69 likes :(

[@AnnoyingMoose]

Thanos is just an imaginary irrational.

[@samgordon9756]

"You call that big? As if." --He Who Remains (Kang)

[@michaeleckert7692]

He really needed somehelp poor guy thought 2× meant something hell he didn't even push an octave

[@hexenex]

Please, don't apologize for the length of the video. It is extremely fascinating and informative. And, infinite, of course!

[@SahilBansal17]

In the proof at 04:22, in the first line, it must be suppose a/b = sqrt(2).

[@ShaneClough]

a/b sqrt(2) = 1 is essentially the same as a/b = sqrt(2), if you divide through by sqrt(2) you get a/b * 1/sqrt(2) = 1. Taking the reciprocal of both sides then leaves you with b/a sqrt(2) = 1, and because a and b are just arbitrary integers, you can use them interchangeably leaving you with a/b sqrt(2) = 1.

[@blazedinfernape886]

@@ShaneClough what are you saying???? The supposition itself is wrong it should be a/b =sqrt2 Not a/b = 2

[@1PUREROSE]

Shane Clough thanks

[@itisdevonly]

Yeah, I got really confused by the proof until I realized the first line was supposed to say sqrt(2) rather than 2.

[@Scubadooper]

Surely it doesn't take much to edit the video and correct the proof. If you've still got it put a square root sign in on the very first line, if not rewrite it!

[@giwrgos_kakep5020]

"You can't really count all the natural numbers, you die at about 3 billion" - instant subscribe

Ah great video! It reminds me the first years of my engineering with calculus and algebra and their demonstrations. I could find the beauty of some demonstrations, and some other nasty ones... There were some challenges of course, but at the end finding the order in the chaos was part of the joy too. Thanks for the explanation, that I'm sure has hard work behind.

[@theultimatereductionist7592]

Order density of a set (between any two x & z in the set there exists a y in the set such that x < y < z) implies automatically an infinite set. But, you need the additional & independent axiom of the least upper bound property to make the set uncountable. Both Q & R have order density but only R has the LUB property.

[@therealEmpyre]

I see an issue with the first proof you showed for real numbers not being enumerable. You said they are not enumerable because there are infinitely many other real numbers between any two real numbers, no matter how close they are. This is true, but the same is true for rational numbers, which have been proven to be enumerable, so that is not a proof that real numbers are non-enumerable. The diagonal proof is convincing, though.

[@terryendicott2939]

Since there is a rational number between any two reals, one can pick two distinct transcendental numbers and put an infinite number of rationals between them. So by the first "proof", does this mean that there are more rationals than transcendentals? The second argument has one small omission. The decimal representation of rational numbers is not necessarily unique. For example .50 = .499999999999999999999999999999999999999999999999999999999999999999.... This actually does not affect her proof, but should be mentioned and shown that it won't.

[@therealEmpyre]

@@terryendicott2939 I didn't say that there is a rational number between any two reals, or at least I didn't mean to say that. There are infinitely many rational numbers between any two rational numbers (not between any 2 real numbers). That makes them very densely packed, and yet I know that the irrationals, and more specifically the transcendentals, are much more densely packed still. I was not arguing against what she was trying to prove, just against that specific proof as she described it.

[@terryendicott2939]

@@therealEmpyre I know you did not say that between any two reals is a rational. I did. If this were not true rulers would not work. I just wanted to point out that the argument used could provide a "proof" that a set with cardinality of the continuum would be "smaller:" than the integers. Just a small variation of your observation.

[@therealEmpyre]

@@terryendicott2939 I think you are wrong about there being a rational number between any 2 reals numbers, but it is just beyond my level of expertise to explain why. I am sorry that I can be of no further help.

[@dan_rust_maths]

@@therealEmpyre Between every two distinct real numbers exist infinitely many rational numbers. Let x < y be two real numbers. Let n be such that (1/10)^n < y-x. Such an n must exist as y-x is strictly positive (and then use Archimedes' property). Now Let z be the number you get by only taking the first n+1 decimal places of y (if y has a trailing sequence of 0s in its decimal representation which begins before the n+1st position, then we can do a similar trick with x by taking the first n+1 decimal places and then adding (1/10)^(n+2) ). It should be clear that z is rational as it has only finitely many non-zero digits. Clearly z is strictly less than y, and by our choice of n, it can easily be shown that z is strictly greater than x. To get infinitely many, just take larger and larger ns in the above argument.

[@prbroussard]

Yeah! Always fun to hear about Aleph null and Aleph one

[@danielspivak3926]

Saying that the size of R is aleph one is the same as the continuum hypothesis. You're thinking of beta null and beta one.

[@juddgoswick2024]

Jade's enthusiasm is transcendental. For a formal proof, watch her videos!

[@SammiCPC79]

I remember back in high school proposing that since there were an infinity of points between any 2 locations, movement should be impossible because each point to point takes some time and therefore any movement should take an infite amount of time. A friend who was a little ahead of me in maths introduced me to the concept of 'infinitely small' or infinitesimal - I can still remember that feeling, like a click in my mind. I'm no math student, I lived the life of a manual labourer - I did learn z80 assembly in my free time as an adult which I still find a lot of fun on emulators. Electronic base 2 math has got some really really neat little tricks. Transcendentals though right? I always tried to visualise what these abstract concepts would mean in real terms - so a number that no matter what scale you use it always falls between your marks. In case of Pi, you might think you could just use a scale of Pi multiples, 0xPi, 1xPi, 2xPi etc. - but in real terms you would never be able to accurately place the first mark after 0 (the scale could possibly exist but not possibly be created). If the scale is based on multiples of a transcendental, every mark on that scale (except 0) is transcendental thus outside the reals and therefore impossible to pin down. I remain in awe of these numbers. Then of course there's imaginary numbers - but trying to imagine what the square root of -1 apples would look like ... that's even worse than the transcendentals! I still hope one day I WILL find my Pi multiple ruler with little wobbly marks that move every time you look closer at them. Or perhaps more likely someone will pop along here to inform me how wrongly I'm thinking about the whole thing and give me another little conceptual nudge like my friend in high school and the infinitesimals that make all movement possible...

[@danksamosa3952]

Zeno’s paradox

[@justanotherdump-ft5rf]

Lol that sounds like zeno

[@linkon_]

Thank you for explaining such a complex topic so easily. It takes some time to grasp the content of the video, but still explained effectively. 🙌👏👍

[@xpkareem]

He didn't drive himself insane thinking about numbers, he was always going to go insane, he just spent his time waiting to go insane thinking about numbers.

[@dru4670]

This sounds more plausible for some reason

[@SpencerTwiddy]

Cool vid as always👍🏼 here's a quick video topic suggestion: the Anthropic Principle. It might be a bit philosophical but I hear about it a lot from leading physicists, and think you would probably have an interesting and unique way of explaining it. Thanks!

[@coreybray9834]

We note first that the squares of N are contained as a proper subset within the set N. So, if N mapps to the squares of N, and the squares of N maps back to the squares of N in N, then the set N must have a Cardinality that is larger than its own Cardinality, a clear contradiction. This arises from the fact that we did not exhaust N in our second mapping, but only exhausted a subset of N. In fact, if we use the standard definition for an infinite set, we run into some serious and blatantly overlooked problems. For all m in Z^+, there exists an element m+1 in Z^+, such that m < m+1 in Z^+. This set Z^+ is endless by construction, because for every m in Z^+ we know we always have one larger value m+1. But, at no time do we everfind, when building Z^+ that the transition from m to m+1 is ever a leap from the finite to the infinite. There is always a finite distance of +1 that exists between all values m and m+1 in Z^+. Consequently, the set is both endless and finite everywhere, not endless and infinite. Anyone doubting this result is encouraged to identify the value for m, such that m+1 is no longer finite. Also, notice that mathematicians argue that infinity = infinity + 1 = infinity + 2 and so on, but if all these steps equal infinity, then the very idea of magnitude, as well as the fundamental theorem of arithmetic, breaks down at infinity. But, the concept of magnitude and the fundamental theorem of arithmetic never breaks down in the case of m < m+1, because m and m+1 cannot possibly be represented as the same unique product of primes to their powers as with the earlier case where all the above forms of infinity equal each other. So, Z^+ is not an infinite set, because all values of m and m+1 in Z^+, such that m => 2, will never have the same magnitude, nor the same unique prime factorization.

[@NotHPotter]

All of set theory's talk of alphas, betas and omegas is Greek to me.

[@jesse2535]

its literaly greek my baby

[@aspiringcloudexpert5127]

@Kat Cut It's a pun on an English idiom.

[@Blox117]

i am the alpha!

[@sharpnova2]

honestly it's clearly greek to her as well. her first proof of the uncountability of the continuum is completely wrong and in fact could apply to the rationals as well which are provably countable. this video is littered with potentially harmful errors and generalizations.

[@BlackJar72]

It could really use some alephs. (Just google "aleph infinity.")

[@omerresnikoff3565]

It's actually even easier than that, since Cantor proved that the power set can't have a bijection (pairing) onto the original set, therefore the power set of the reals is even a bigger number. More complex mathematics as the use of the von Neumann-Bernays-Gödel (NBG) set theory allows you to see that the cardinal of the reals is the next infinite cardinal after the enumerable, that is, there is no posible intermedium cardinal; this is called the continuum hypothesis and Gödel has a great article concerning it and the uses of the axiom of choice (AC)

[@haisontrinh727]

I'm glad that you made a video about Cantor's work, he is one of my idols, thank you so much. However, you forgot to mention the relation between cardinality of natural numbers and real numbers (c=2^ℵ0) and its proof, the Continuum hypothesis, which drove him insane. These are the real fun parts (except for insane part, poor Cantor!).

[@wienerfotograf]

Cantor was wrong, in my opinion all infinities are the same. I can prove it if you want.

[@coldblaze100]

This is the kind of shit Dumbledore was probably working on. Keep it up

[@GMPStudios]

And this my folks, is what you need to know to understand limits in calculus.

[@NotHPotter]

Hehe, and the limits of calculus.

[@stevec7819]

@@NotHPotter Weierstrass function

[@NotHPotter]

@@stevec7819 well played

[@GMPStudios]

@Ian M But DC has a lot of limits xD

[@hybmnzz2658]

Hell no. That is so different. Limits are about the arbitrarily large. Whether you are talking formally or informally.

[@rish1459]

Another lovely video. I love how you show a proof by induction without getting into the guts of how to prove by induction! Brilliant.

[@AlexKnauth]

Wait the “proof by continuum” thing at 7:43 : why doesn’t that also imply that the rational numbers are non-enumerable? If the rational numbers are enumerable, and this proof technique seems to contradict that, is the proof technique invalid? Or, does it not actually apply to the rationals because of something I’m not seeing?

[@JiveDadson]

the diagonalized number is not rational.

[@AlexKnauth]

@@JiveDadson Yes, I know. That's how I know the Diagonalization proof doesn't work for the Rationals. But what about the "continuum" proof? Does that rely on any properties of the Reals that Rationals don't also share?

[@stevethecatcouch6532]

@@AlexKnauth The continuum proof is not sound. Her assertion that you get stuck at alpha(v) and beta(v) is unsupported. Nothing about her list of omegas prevents finding a number on the list that falls between alpha(v) and beta(v).

[@DarwinsStepChildren]

Her written explanation doesn't explain the "proof by continuum" very well. Even using the phrase Real Numbers is a bit confusing, because Real numbers include natural, integers, and rational numbers, AS WELL AS, irrational and transcendental. The proof by continuum only applies to the irrational and transcendental numbers. Technically, she should have said "irrational & transcendental" in place of "Real Numbers". Also a bit confusing is that the proof by continuum explains something intuitive. Some scientific proofs explain something(s) that is/are new, and unknown through experience. Take the age of the earth. It doesn't matter how much experience one has on earth, the age of the earth is not intuitive, in fact, the only way Homo sapiens can understand the age of the earth is via the scientific proof. However, take something like gravity, it is intuitive to Homo sapiens, as well as some other species, that if you drop an object, it will fall towards the center of the earth. This intuition comes with experience. Most Homo sapiens, when shown Newton's or Einstein's equation for gravity, after asking what it explains and being told it explains why a stone thrown in the air will eventually hit the ground, will say they already knew this and didn't need a mathematical equation to tell them. Even given this, gravity can't be explained through intuition, and requires mathematical proof to "exist", or be used in mathematical equations. The proof by continuum is similar. The important aspect for ANY infinite set is whether or not it can be paired with the Natural Numbers. Given alpha and beta, the fact that there is an infinite number of numbers between the two is irrelevant. Take alpha to be zero, and beta to be one. There are an infinite number of rational numbers between them. That isn't important. What is important is whether they can be matched with another Natural Number. There is 1/10, 1/100, 1/1000, etc. between 0 and 1; however, 1/10 "could" be matched with 10. 1/100 could be matched with 100, and 1/1000 could be matched with 1000, and so on and so forth. The important aspect of proof by continuum is that the irrational and transcendental numbers cannot be matched with the Natural numbers. The video's author does say this ~ 8:50. Take pi, a transcendental number, what natural number pairs with pi? None, again this is intuitive because the number goes on forever with no repeating pattern. NOW, PROVE Pi HAS NO NATURAL NUMBER TO PAIR WITH. This is where the proof by continuum comes into play. Take alpha to equal Pi, and take Beta to equal Pi plus one. There will be an infinite number of numbers between Pi and Pi plus one that CANNOT BE PAIRED WITH ANY Natural number, and it is this fact that PROVES that Pi is an irrational/transcendental number, and cannot be enumerated, and is therefore contained in a "different" infinity than the enumerated natural, integer, and rational numbers.

[@meetjoshi9853]

@@DarwinsStepChildren Thank you so much man. You cleared all of my doubts

[@dcryan]

Loving your videos! 👏 You really know how to break a topic down into bite size pieces. You‘re able to take comprehensive topics and make them more approachable and less daunting.

[@TheGrooseIsLoose]

I’ve heard this topic explained twice before: once as a guest lecture in a class whose purpose was to fill in all the gaps that weren’t covered in any other classes in the math degree, and once from some professors from the math department at my school giving a talk at one of their colloquia about a paper they had recently published that required some background on this. Most of the other math professors in the room were having trouble understanding a lot of it. In this video you covered maybe a third to half of what they covered on the topic in like a quarter of the time, and explained it just as well if not better. This kind of stuff needs to be on KZbin more. A lot of educational videos cover very elementary topics at a surface level, which are great for getting kids interested in the field, but don’t add a lot in terms of freely available information for people who actually want to learn something outside of school. There are also long, in-depth lessons on a topic, which are also essential for a full understanding, but often favor completeness of information over providing a general understanding. Things in the middle, like this video, that introduce more advanced topics and provide a basis of knowledge for people to either watch longer videos or read about a subject are essential to helping people find and learn about topics outside of school. One of the interesting things about this topic in specific is that it requires almost no prior knowledge of math (as in you need only a surface level knowledge set theory and number theory) to start to understand it, which can lead people without a strong math background to get interested and start reaching out into the other topics to be able to answer questions they may have.

[@upandatom]

Hi "The Austin". Thank you so much for this comment, it was probably my favorite I've read for a while. Truly brightened my day :) I'm glad you were able to get something out of it. And yeah it's so cool how you really don't need much background information at all to understand something so profound. That's the beauty of set theory.

[@wienerfotograf]

Yeah, it is really beautiful but it is unfortunately a wrong theory, when it comes to the „different“ infinities. I can prove it here if you want.

[@perrzzz]

Wow, yours presenting is beyond amazing!

[@russellcannon9194]

Wow. Wow! WOW! Very well done. Thank you so much for making this video. I have known for a long time that some infinities are larger than others, but whenever I tried to explain this to my skeptical friends, I would fail miserably. The explanation you provide here is so elegant and succinct that all I need do is recommend this video to my friends as THE proof. Thank you so very much. A true measure of genius is not in what one knows but in the ability one possesses to successfully explain a complex concept in a way that can be apprehended by those who struggle with it. You are a genius. Vey well done. Cheers, Russ

[@aspiringcloudexpert5127]

Yay, a new Up and Atom video!

[@garybrisebois2667]

Thanks for posting this Jade. I have loved this subject since I first studied it almost forty years ago. I learned that the Real numbers were uncountable (the term we used) and therefore a "stronger infinity" than the natural numbers. I had never realized that the transcendentals were the downfall of the countability of the Reals, I thought it was all the Irrationals. I would love to see the proof regarding that! Anyhoo recently, in the past few years, I looked into Strengths of Infinitude further, and learned that with dimensionality you can find infinitudes stronger than the Reals: the number of line segments on a line is Stronger than Reals, the number of curves on a plane is still stronger than that, the number of shapes in 3-space still stronger, and so on! This means that there is AN INFINITUDE of INFINITY STRENGTHS! Blew my mind. Thanks again, came here from Tom Scott :)

[@wienerfotograf]

So you could also say that Cantor was wrong and all infinities are the same. They are Infinite!

[@equesdeventusoccasus]

Excellent work. It feels like this could be the start to a new series you could post along side your other videos as time permits. I'd be happy if it was.

[@kylorenkardashian5518]

nice double infinitys

[@markhaus]

Reading about mathematicians you begin to wonder if going too deep into maths awakens the “Great Old Ones” and drives you insane

[@soutrikgun9133]

I love the craft of the comment

[@Raison_d-etre]

No.

[@reikiorgone]

only awakens one to their spirit essence... the singularity

[@Raison_d-etre]

@@reikiorgone No need for that hypothesis.

[@reikiorgone]

@@Raison_d-etre that's an a priori assumption... Love ya!

[@SuperRousku]

Your result at 8:25 is also true for rationals. There is no finite-sized span that doesn't contain an infine number of fractionals, yet they are just as enumerable as the natural numbers. How cool is that.

[@zbyszekz77]

Great video, maybe part II about continuum hypothesis and Godel incompleteness theorem?

[@upandatom]

yes i so want to cover the incompleteness theorem. just gotta wrap my head around it first

[@TheDecguy]

I was waiting for Godel to show up.

[@omerresnikoff3565]

Which one of Gödel's three incompleteness theorems?

[@martinkunev9911]

8:40 there are an infinity of numbers between alpha and beta but this doesn't mean it's a continuum. the same argument applies for rational numbers (we say the rationals are dense).

[@KaiHenningsen]

And of course, given any (infinite or not) set, you can *always* create a set with a larger cardinality: just look at the set whose elements are all the possible subsets of the first set. (In the finite case, if the first set had the cardinality C, the second would have the cardinality 2^C. In the infinite case, if you start with the cardinality of the natural numbers, you get the cardinality of the reals. If you start with those ... you get the point.) Of course, that rises an interesting question: can there be a set with a cardinality _between_ the naturals and the reals?

[@MikeRosoftJH]

Of course, the answer is that set theory (without additional axioms) can't settle the question in the positive or negative. And it's even worse in absence of axiom of choice. It can be proven that for every well-ordered set there's another well-ordered set with strictly greater cardinality. But without axiom of choice you can't prove that real numbers can be well-ordered. (And even if there's no well-ordering relation on reals, it's consistent that at the same time there is no set whose cardinality is strictly between the natural numbers and the real numbers.)

[@kaushikgupta9490]

4:40 Hippasus, is that really you ? 😂😂😂

[@thedrewsky4915]

recently I was looking at what concepts and basic theorems I would need to learn if I were to take a certain major for college and cantor's principle was listed under it. I watched three videos before finding this one and left them completely clueless with what the heck cantors theory was even remotely about. Thank you so much for explaining it, now I at least know what I am getting myself into.

[@bernard2735]

Actually Euclid uses the word ‘ἄλογος’ (alogos - literally ‘irrational) in The Elements.

[@KhAnubis]

Not sure if you saw it or not, but a few months ago on Twitter, I posted something asking if an infinite stack of €20 bills would be worth the same as an infinite stack of €5 bills. Most people said 'yes', but one person pointed out how if you wanted to withdraw the money, it would be faster to do so from the €20 stack. Also yes, I did notice the world map at the back of your room.

[@upandatom]

yeah it would be the same they were right :)

[@cakeisamadeupdrug6134]

0.00000000000...1 == 0 Hehe I usually teach the sum of an infinite series via the medium of Harry Potter tbh. Assuming each Horcrux splits Voldemort's soul perfectly in half, obviously.

[@alefdias4468]

Hahaha, I think I will use it too in calculus 2 to show intuitively the convergence of infinite series, seems much more interesting to divide a soul than a square...

[@superfluidity]

Not really. '0.00000000000...1' is just a meaningless string of symbols. It doesn't represent any number. You can't have an infinite string of zeroes and then a 1, since there's no end to the infinite string and so nowhere to put the 1.

[@alefdias4468]

@@superfluidity I think math is more about the abstractions we make, not about what kind of strings of symbols represent what, I see the "infinite" zeros after the comma as the idea of approaching a limit, the more zeros you put in between, the closer you get to zero, so, if to put a "methematical rigorosity", we can use the accepted limit version, is the same as saying that, as n goes to infinity, Lim [1/10^n]=0. To get more geometrical, it is about the number line, frontiers between numbers get really blurry when the continuous real line is explored... Many funny things happen when exploring the in-between of the numbers, the irrational and the transcendental ones are really interesting...

[@cezarcatalin1406]

If the number of 0 is a quasi-infinite number, you just described the first rational number above 0.

[@FactoryofRedstone]

@@cezarcatalin1406 If you take lim(n->infinity)(1/10^n) you don't get a number above 0 you get 0. And what it means that they are infinite non-enumerable is, that there is no 1st number above 0, or above anything. For saying there is a 1st number, you need to be able to count (list) them, but there isn't.

[@impowerzone]

Your expiation of the topic was amazing. I was struggling with some of the concepts while learning discrete mathematics for my data science course but you made it fun and easy to understand, please keep going your work is useful in many ways :) thanks for making the video.

[@MisterrLi]

Thanks for your little introduction on infinity. It is hardly a complete one, however, the nature of infinity is a little bit richer than this. When talking about infinite sets mathematically, there are many ways to express them. Cantor created one of the methods, in which he equals all infinite sets that are listable. It is however easy to see that therer are other possible ways to do it. You can treat all infinite sets together as a whole (as one concept, traditionally "∞"), where every infinite set is bigger than every finite set, and that's probably the most common and simple definition of an infinite set used in real analysis. Then there are the Cantorian hierarchy of transfinites, starting with finite numbers, then countable infinite numbers, then bigger and bigger levels of infinite sets, one level constructed from the smaller one below it. Yet another definition of infinite sets actually separates different countable infinities, like the set of all natural numbers and the set of all natural number squares. Nothing is wrong with Cantor's system, there are just many different ways to express infinite sets and their relations, and some ways do it more precise than Cantor did. The pairing up method doesn't work for this more precise definition of infinity, so instead nonstandard analysis numbers are used. These numbers have better precision compared to the usual real numbers and can also express infinite sets, so they make it easier to express and compare infinite sizes of sets. What system to use depends (as always) on your need; how much precision that is asked for and which tool will work best on the task at hand, not which system that is the best in every situation.

[@jeromedavies2408]

I never liked Cantors diagonal proof and didn't really get this until I saw Proofs about enumeration. The question "what's the first number after zero?" got me.

[@MikeRosoftJH]

But that doesn't really prove that real numbers are uncountable. Yes, real numbers are a dense set - there are no adjacent real numbers: between any two real numbers x and y there is another distinct from the two, such as (x+y)/2. But rational numbers have the same property, and they are countably infinite. To prove that real numbers are really uncountable means to show that no infinite sequence of real numbers (no function from natural numbers to real numbers) can cover all reals. The diagonal proof is one way to do this; there are other proofs.

[@alefdias4468]

I love how passionately you talk about how exciting number theory and set theory are... I am graduating in engineering and people don't really see the beauty in math, just its usefulness, which makes me a little sad when helping people understand, let's say, calculus and the amazing things that make my eyes shine do not excite them in any way

[@upandatom]

I know right?! I think it's because it's very abstract and when we learn about it its uses aren't emphasized. Oh well :(

[@alefdias4468]

@@upandatom , exactly! I really enjoy talking about math, I like it not only for how useful it can be, I just find so much beauty in it! So much pleasure in simply understanding and visualizing the concepts, it is simply amazing! I wish I could show math to other people through my eyes... It is so hard to explain why you find so much beauty in something(showing people things they don't see is probably hard in any situation). I really enjoy learning and understanding things, and the ones I like the most are in math... So general, so logic, like e and π transcend our meanings based in our daily uses of numbers, math transcends our reality!(and can still predict things about it, even abstractly on a completely different dimension, sometimes literally).

[@markgraham2312]

I wish you could have been around in 1978 and spoke to my RA in college who insisted that there weren't different levels of infinity. I would have to wait until my Junior year to learn about the countability of sets. Also, instead of saying: "I physically could not not include stuff," say: "I physically could not omit stuff." Great job!

[@AyushGupta-rb4hb]

"Some infinities are bigger than other infinities"- Fault in our stars

[@carlhopkinson]

I took Set Theory in college and it was the most mind warping course I ever didn't drop.

[@JakubH]

doesn't the argument starting at 7:42 hold also for just the rationals?

[@iVideoCommenter]

No, because a rational is defined by the division of two integers, which are defined by finite successors of 0.

[@Eulercrosser]

Yes, you are correct. The argument breaks down at 8:09 when she says "suppose you get stuck" and assumes that you will get stuck without ever proving it (which cannot be proved when you've assumed that you have a complete list).

[@AlexKnauth]

Jakub Homola +1 You’re talking about the “proof by continuum” thing at 7:43, right? I have the same question. Is there something I/we’re missing that makes it apply to the reals but not the rationals?

[@JakubH]

@@Eulercrosser about that "get stuck", I think she assumed that for contradiction to prove that you can always find numbers between a' and b'. She than found other numbers in between, creating that contradiction.

[@JakubH]

@@AlexKnauth yeah, she later said "this is the infinity of the continuum", but that doesn't seem right to me, since that argument also applies for rationals

[@ronmasters751]

I like the graphic proof of the countable Rationals. Integers on the X & Y axes. Every rational number corresponds to an ordered pair. Start at (0,0), and count, spiraling outwards to (1,0), (1,1), (0,1), (-1,1), ...

[@JJ-kl7eq]

The SPCA is going to be interested in your sets where you draw lines between the forks and the dogs. Is there a dark side to you we need to know about?

[@Blox117]

whats wrong with eating dogs?

[@robinw77]

@@Blox117 there's nothing necessarily wrong with that, but sticking forks in living dogs is cruel :-)

[@fakegay68]

I hope it has nothing to do with hunger.

[@chrisg3030]

When you write the infinity symbol (something like "oo") you start an endless count 1 2 3 ... that never stops, even if you go away and forget about it. It follows that oo > oo since you started the endless count on the left earlier than the one on the right so it already got to a bigger number. That's why infinity comes in different sizes.

[@chrisg3030]

@BlueBoy 1 Well you can rule out time playing a part in a paradox if you like, even though it clearly does play a part in all our actions, including mathematical operations, which take place one after the other rather than exist all together timelessly and simultaneously. The latter view was only appropriate when math and science were considered the contemplation of the eternal. But I think ruling out the time factor is to rule out the possibility of demystification.

[@chrisg3030]

@BlueBoy 1 But that's just the distinction I thought I was making. Anyway yes, I guess you could say that about any number. It doesn't just pop into existence when you reach it by counting. I don't have to count to 257 to know it's already there, I can just name it by constructing it out of a set of single digits, or do a calculation like 199 + 58. Right?

[@mrnarason]

"a set is a group" lol There is a reason mathematicians use the specfic wordy definition, "a set is a collection of elements".

[@heimdall1973]

Describing a set as a group, collection, whatever else you can think of, is just playing with words, not defining a set. Mathematicians *do not* define a set. They describe sets by telling us their properties and properties of set membership. (A few such properties are given as axioms, others are proven.) So a set is just an abstract object. For example, having an object x, there is a unique "set", let's call it A, such that x is "a member of" A and any object that is not x is not "a member of" A. Such "set" we also describe as {x}, but {x} is a distinct object from x. By the way, a group is another mathematical term (it describes a set with a binary operation defined on it, satisfying certain properties).

[@pieffe8]

Thanks for this video. Yes I was confused by your first proof of the uncountability of the Real numbers. However I have a small problem with the second one too. The problems is because some real numbers have two decimal representations. For instance, 0.1 and 0.0999.... . So there may be a chance that the new real number generated from the diagonal may occur on the list in the other form. I've been told there are many ways of dealing with it. One way I've discovered is by adding 2 to each digit instead of 1. This guarantee that the real number is indeed different. Continue doing the great job!

[@wienerfotograf]

I can prove that Cantor was wrong and every infinity is the same size. Set of R = Set of N.

[@Msaber2012]

Number theory is infinitely interesting 🙃

[@maxwellsequation4887]

Ikr

[@saichaitanya7060]

hi brother ,what is difference between set theory and numbere theory?

[@dnlwood32]

Absolutely love the enthusiasm. I watch so many science videos and enthusiasm certainly makes them more watchable.

[@GMPStudios]

Infinity War!

[@pfefferfilm]

"He spent the last few years of his life in and out of mental institutions. Today, we know him for his work on numbers" Beatiful

[@geo3043]

I didn't understand the first proof: why can't you do the same thing with fractions?

[@AlexKnauth]

geo froid You’re talking about the “proof by continuum” thing at 7:43, right? I have the same question. Is there something I/we’re missing that makes it apply to the reals but not the rationals?

[@pierreabbat6157]

You'll get an irrational number as the limit of the two sequences.

[@geo3043]

The problem for me is that it seems to be supposed that there is a finite quantity of numbers (in the list of omegas) between alpha and beta. But that's not true, even whith only fractions. You can always find an infinity of fractions between two numbers, no matter how close they are. So, with the infinite list of fractions, you would always be able to take a closer number of alpha by going further on the list. Therefore, the closest number of alpha does not really exist on the list.

[@geo3043]

@@hojasrayadas thank's for the answer, i'll have to look at the way this list is constructed to fully understand.

[@mitjamastnak9206]

Sorry for my previous comment about the first proof - I see that my concern was already addressed in the comments years ago. Great job!

[@taht-qrm6696]

8:41 what i drew on my math exam paper

[@mepaulb]

I love the drawings. I learned Georg Cantor, after thinking so much about infinity, became an evil wizard.

[@quahntasy]

Who is here before 100 views? You sir are greater than infinity.

[@garyk3478]

I am! After I discard the other 2,106 views as outliers, anyway.

[@hughbarton775]

I am, in comparison to the great majority of your correspondents, almost comically undereducated....but I think I have, thanks to your (seemingly!) lucid explanations of set theory, had a weird realization....it would seem that several things which I "knew" are oddly related: (1) I first learned of Bernoulli working on cars(!)...a crude understanding of his principle involving behavior of fluids was crucial to modifying and improving automobile engines. (2) Apparently, his work involving analysis of the math of compound interest was a bedrock of the math utilized in calculating interest! (yea, I sold cars too....) And, finally, his involvement in set theory has really helped me(hopefully!) understand infinity a little better.... This has helped me to decide , after considerable musing on the question posited by Ms.(Dr.? Professor? Dunno....) Hannah Fry regarding the notion: "Is math inherent or created by our perception of the universe?" At this moment, I am going with inherent... P.S. : apologizes in advance to all you real mathematicians who by now are probably thinking, " Who is this nitwit? "

[@neurodivergent4life]

Jade's friend recieves a call: - Hey! Guess what! My infinite is bigger than yours!! ........ Ha!! - *Sigh*... It's 3am, Jade, go to bed...

[@upandatom]

how did you know about this conversation?

[@neurodivergent4life]

@@upandatom "I was there, I’m everywhere. Isn’t it beautiful world when everyone lives together, maybe it’s just me. Maybe it is I who lives with everyone. You are never alone. My name is... ... Alexa!" Happy Halloween. Lol

[@thatellipsisguy8984]

And that folks, is why the French husband is tired...

[@drewtheunspoken3988]

Sometimes, as I lie awake at night, I like to contemplate infinity. I stop when it feels like my mind is bending.

[@Dudleymiddleton]

You make brain-melting mathematics sooo sweet! :)

[@pranaynarvariya8342]

After class, this video has cleared some doubts my doubts. Thanks a lot.😊

[@spyrex3988]

The amount of time I've studied set theory throughout school and college is infinitely large

[@brandonnicholson2633]

The idea that one infinity can be bigger than another is built on the mistaken idea that infinity is a number, and therefore, has a size and can be measured. But it's not a number at all, so we shouldn't assume that we can treat it as one. Infinity breaks a lot of basic mathematical laws, because it's not a number, and so it isn't bound by those laws. Infinity doesn't belong to the set of all numbers so Cantor's theory has a very fundamental flaw.

[@Lonly82Wolf]

Yes, only a closed set can be compared to another closed set to see if one is bigger than the other, but when talking about infinity it is not a closed set it goes on forever, because of this infinity can't be bigger or smaller than infinity, if they are both infinity then they are the same size infinite size.

[@gidi5779]

Infinity is not assumed to be a number and it surely is not treated that way. When one says "one set is bigger than the other" they mean there exists a surjection from it to the other. Also, I am not aware of any "mathematical law" that is "broken" by infinity

[@gidi5779]

@@Lonly82Wolf What do you mean by closed set?

[@Lonly82Wolf]

@@gidi5779 I mean a closed group of numbers, that you can count.

[@Lonly82Wolf]

@@gidi5779 Although it seems logical the idea to compare infinits, by comparing their parts they go on for ever, if they have no end they are the same size infinite.

[@englishiseasy6092]

I love the way you explain the most complex and even boring things ever and I keep watching it over and over and over again. ❤❤❤

[@matthewalan59]

You are infinitely adorable. I guess I am going to have watch all your videos.

[@joelrivardguitar]

Good video. If I remember correctly the first infinity is w (omega) or Aleph-null/Aleph 0 and the next one is Aleph 1. Infinity and the Mind by Rudy Rucker is one of my favorite books ever.

[@gJonii]

Your first proof for non-enumerability of reals seems to work equally well on rational numbers. Meaning, it's wrong

[@MikeRosoftJH]

The proof is incomplete as stated. There are actually three different cases: a) the two sequences a(n) and b(n) are finite; b) the sequences are infinite and have the same limit; c) the sequences are infinite and have a different limit. In either case there is some real number not in the sequence. (For details, see the linked video in the pinned comment, or the Wikipedia article "Georg Cantor's first set theory article".) You can apply the proof (or the better-known diagonal proof) to a sequence containing all rational numbers, but the resulting number must be irrational.

[@Lotschi]

I just discovered your channel at midnight. Yayy, another excuse to procrastinate...

[@nnfefe9451]

Kudos to you for you have a command of what you are talking about.

[@RantsofaDrunkGuy]

There is that line by Matt Parker that really screwed up with my head, he was talking about non computable numbers and his conclusion was: "We as mathematicians like to think we know numbers, when in fact we have found none of them"

[@TheHernanNoguera]

"Yeah, the Greeks didn’t like irrationals" **throw a man off a cliff because, you know... numbers**

[@darkwater234]

Well played! Greeks were suppose to be the architects of reason and wisdom and yet they rage quit math and killed some poor dude.

[@CarlosGonzalez-rg6ht]

Absolutely fantastic video! Thanks very much.

[@keniangervo8417]

This was better explained than from a teacher

[@richardaversa7128]

She is a teacher! In fact her audience of students is probably larger than any college professor or high school teacher's.

[@DanielOlaiDanielsen]

I'm a simple guy, I see a cute face on a thumbnail and I click it. But woah, you're as good of a teacher as any of the numberphile ones, the way you explain it doesn't make my head hurt. Wish my maths teachers had been like that, I wouldn't have given up on the subject in 3rd grade. Now off to see the rest of the videos!

[@W4LL37SK83R]

In a college course I took (I think it was linear systems) I remember talking about the "weights" of infinity. That's probably not a mathematical term, but just an analogy the professor used. It was in the context of the dirac delta function, in which y=infinity where x=0, but y=0 everywhere else. It is the derivative of the heaviside step function in which y is 0 for all negative x values, and 1 for all positive x values. When we were talking about the "weight" of an infinite value, in this context it referred to the y value at x=0 for some function that was a multiple of the dirac delta function. So while 1*diracDelta(0) = infinity and 2*diracDelta(0) = infinity (which might make it appear as though 1*diracDelta() = 2*diracDelta(), since y=0 for both functions when x=/=0 ), the two are actually not equal because the integrals of the 2 functions are equal to different heaviside functions. 1*heaviside() =/= 2*heaviside() because the value of y when x>0 is different. By that logic we would say that the "weights" of 1*diracDelta(0) and 2*diracDelta(0) are different. This would probably make more sense in a drawing than a comment, but the point I am getting at is that even if 2 infinities have the same cardinality, that doesn't always mean they are the same "size" (though my definition of size here is probably different than the one in the video). I might be wrong, but it seems based on this video that 1*diracDelta(0) and 2*diracDelta(0) are equal to infinities that are cardinally equal to each other. But they still must be unequal in some way since they have different integrals. The way we described this difference in the linear systems course I took was by saying the infinities had different "weights". So if all of that made sense, I'd love to see a video that talks about this aspect of infinity. I've looked around for a better mathematical term for what I've been calling the "weight" of an infinity, but I haven't been able to find one. Maybe if you know or could find the right term, you could make a video out of it?

[@iuyesup991]

沙比

[@picodev5597]

well done. about the cantor's diagonal it's not correct to "add 1" becaus if you add the carries to the digit to the left you' wont be sure to not get a precedent nimber. Usually it's easier to create the new number by using 8 if the original digit is not 8 and 1 instead. Any digit but 9 could be used ti avoid the .99999=1 problem.

[@zeon137]

Not really, it's actually any operation mod9

[@pierreabbat6157]

I'd add 5 mod 10 (or 2 mod 4 etc.; as long as the base is at least 4, it works).

[@dru4670]

Started from your recent video then came back to watch this. Really love it. Am a sophomore engineering student 🤗.

[@elyepes19]

Thanks for the video and the topic! A great intro to it is on the book "A Journey Through Genius" by W. Dunham, over there there are other ideas like quadrature of circle (impossible because of transcendental numbers), also the summation of 1/(n^2) = (pi^2)/6 (why pi, a transcendental, here?) with n going to infinity, and many more related to this video, and in line with the spirit of the Up and Atom channel

[@peternicholas4750]

The medium article 'Cantor’s Approach to Mathematical Infinity' copies the content and text of your video almost word for word

[@MuffinsAPlenty]

Wow, that's an astonishingly blatant example of plagiarism. Even many of the diagrams used in the "article" as ripped directly from this video. It makes you wonder just how many articles Waldo Otis has stolen from other people.

[@Darkness2179]

Wow. Intelligence AND looks. What a channel to stumble upon! Subscribed!

[@vitaliset]

the funny part about this video is that i coincidentally saw that right before i study about infinity for my exams. Really helped to sort out some shenanigans that was going on that chapter of our course

[@billf7585]

I'm fascinated by this topic, the cardinality of the continuum, and all the strangeness it leads to. Its relatively easy to prove that any open interval has the same cardinality as all real numbers. Another interesting related topic is the Continuum Hypothesis, also advanced by Georg Cantor, who believed it to be true. It states that there is no set whose cardinality is strictly between that of the natural numbers and that of the reals.

[@wilhelmtaylor9863]

When I was in high school I learned of something called "hyper infinity". Example: there are an infinite number of points in a given volume but for every point there are an infinite number of lines passing through it. So logically the number of lines would be larger than the number of points.

[@MikeRosoftJH]

The number of lines is in fact the same as the number of points. Any point is a sequence of three coordinates in real numbers: [x,y,z]. What about lines? A line has an origin and a direction. Both of these are vectors of three coordinates. (But some of these define the same line: If you multiply a direction vector by a real constant, you get the same direction; and likewise you get the same line regardless of the choice of the starting point along the same line.) So to wrap this up: There are R^3-many points in the real space (R being the set of all natural numbers), and no more than R^6-many lines. But there's the following theorem: For any finite number of dimensions n, the sets R^n - the n-dimensional real space - have all the same cardinality (all those sets can be mapped one-to-one). The same is true for any 1-dimensional interval, 2-dimensional shape, 3-dimensional solid body, and so on. I'll show this to you in two dimensions. First: the real line has the same cardinality as an interval on reals. For example, the interval from -pi/2 to pi-2 can be mapped one-to-one with the reals using the tangent function. (Likewise, any two intervals can be mapped one-to-one regardless of length - just use the function x -> n*x for a suitable constant n.) So now take an interval from 0 to 1. Each number in this interval has a decimal expansion: 0.abcdefgh... . Split it to two numbers: 0.aceg... and 0.bdfh... . This uniquely maps a decimal expansion to a pair of decimal expansions (and vice versa), and the set of all numbers in an interval from 0 to 1, and the set of all pairs of numbers from the same interval, have the same cardinality. (There's a technical catch here, which needs to be taken care of in a formal proof. See if you can spot it!) Finally, if an interval (0,1) has the same cardinality as the reals, then the set of pairs of numbers from the same interval has the same cardinality as the set of pairs of reals (which is exactly the 2-dimensional real plane). A useful theorem when dealing with the cardinality of infinite sets is the following: If set X can be mapped one-to-one with a subset of set Y, and Y can be mapped one-to-one with a subset of X, then there also exists a one-to-one function between the two sets (and the sets, by definition, have the same cardinality).

[@jonathanaragones6888]

Another great presentation. Just for a thought, all type of infinity is type of loop. In counting characteristics, what numbers you want to be in the loop? Addition and Multiplication, but those numbers that outside the loop are in Subtraction and Division. At these characteristics you can then define the Finite and Infinite groups. Will these group can be combined? Yes, If the numbers are in the existence and not in the imaginary zone. Then what will happen when you combine them? it will just become one Infinite Numbers as Finite numbers will become Infinite Numbers. But the character of the Infinity loop will change according to the set of rules affecting the set of Infinity, this sometimes called Infinite Dimensional Sets.

[@nickyork8901]

Brilliant explanation, so clear - and entertaining at the same time.

[@quentinludwigs]

I hate math and numbers more than anything in the world....but I can't stop watching your channel! Thank you for making numbers and concepts palatable!!!!

[@kaiblack4489]

Here's the list of mistakes I spotted: - His name is pronounced "gayorg", *not* "jorj". - 4:25 minor point: irrational numbers can have patterns, they just can't have repeating patterns. My favorite example is 0.11010001...., i.e. the number between 0 and 1 with 1s at every position that's a power of 2, and 0s everywhere else. In fact, this number is transcendental! - Plenty of other people have mentioned it I think, but your proof that there are more reals than naturals doesn't work at all. You can do exactly the same process with the rationals instead of the reals and you still arrive at the same supposed "contradiction", even though |Q|=|N|. - 8:54 not technically a mistake, but there are more than 2 types of infinity. As Cantor showed, the size of any set is strictly less than the size of its power set, so you can keep using power sets to generate larger and larger infinities. - 10:11 another minor point: you need to be careful here, because you've forgotten to account for the fact that a single real number can have multiple different decimal expansions. For example, 0.1 = 0.0999999.... To solve this, you need to replace 9s with a digit that isn't 0. 11:10: again not a mistake, just a fun fact: that actually goes for every pair of infinities! If K is any infinite cardinal, then K+K = K*K = K, so in a way that means that K is "infinitely smaller" than the next infinity, and any that come after it. I'm not trying to be mean, I really like your videos! I just thought these things were worth mentioning.

[@deletedaxiom6057]

Where were you 3 years ago? When I learnt this I was a bit confused but fascinated. Nice video, clear, to the point and entertaining.

[@rkara2]

@9.06 ... So, in order to enumerate something you have to start with a non-enumerable source? That happens to be the definition of a computer or Turing machine. "If a computing machine never writes down more than a finite number of symbols of the first kind, it will be called circular. Otherwise it is said to be circle-free." ~ Alan Turing !DA

[@Tomken8d2]

A circle is perhaps the simplest representation of infinity. So is my mouth when you get me started on a topic I find interesting but most others don't. :)

[@Hecatonicosachoron]

Transfinite arithmetic is always such a fun topic! It's nice in the way it subverts intuitions about ordinary arithmetic. Is the argument shown at 7:45 a good proof though? There are infinite rationals to be found between any two different rationals after all. There has to be some distinction between numbers that can be defined by one type of operations and those that cannot; e.g. it is possible to organise all reals in an infinite binary tree - the cardinality of this tree (which does not have the topology of the continuum?) is the same as the cardinality of the continuum. But that doesn't mean that there is a way to calculate every one! Another distinction in the transcendentals would be the numbers that can have a finitely long algorithm to calculate them, such as pi, e, all expressed as b^a where "b" is a rational number and "a" is an algebraic irrational, and those that do not... the former are still countable. I also thought that there is some debate about Hippasus drowning... that he revealed the method of inscribing a dodecahedron on a sphere. And that has more to do with him talking freely about Pythagorean teachings, because they were very secretive. They were so secretive that they put the non-mathematical teachings into layered metaphors that seem nonsensical if you are not told how to interpret them and where also very selective on who could attend the lectures and at what level of detail. They were something between an academy, a monastery and an ancient mystery cult infuenced by Orphism.

[@AlexKnauth]

(Re: Is the argument shown at 7:45 ...) +1, I have the same question. I don't know if there's anything in that proof that would make it apply to the Reals but not the Rationals. If the Rationals are enumerable and this proof technique seems to contradict that, there's probably something wrong with it. Maybe I'm missing something, some property of Reals that this proof uses that makes it not apply to Rationals?

[@Hecatonicosachoron]

I do think there is a similar argument for reals, but instead of picking two numbers, you make two Dedekind cuts (roughly speaking you sort rationals of an infinite sequence of rationals that converges on a real number, which is a way to define the reals and also show that they are distinct from the rationals), so there's a clear way to adapt it for the reals... but still I am not certain it tells us something about cardinality directly.

[@MedlifeCrisis]

You so smart. Someone said something about 'which infinity?' to me the other and made me feel foolish for not knowing. But now I do! Hooray.