Cantor's Infinity Paradox | Set Theory
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- Опубліковано 5 чер 2024
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Cantor sets and the nature of infinity in set theory.
Hi! I'm Jade. Subscribe to Up and Atom for new physics, math and computer science videos every two weeks!
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Sources
The Annotated Turing - Charles Petzold
universalflowuniversity.com/B...
www.amazon.com/Annotated-Turi...
Introducing Infinity, A Graphic Guide - Brian Clegg
www.amazon.com/Introducing-In...
en.wikipedia.org/wiki/Real_an... - Наука та технологія
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. ua-cam.com/video/_qyDBUpAGuo/v-deo.html
Up and Atom Mathemtics PhD student who has taken axiomatic set theory. Lots that needs to be corrected in your video.
oh, now it makes sense
Jade, do you think there will be a Theory of Everything?
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 😅.
@@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.
I love how goofy your drawings are. They look very endearing.
@Joe Tonner amen to that...
I love here drawings. It takes a pure soul to draw like that. I hope she uses them in all of here video 😍
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".
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.”
@@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.
@@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.”
@@savagenovelist2983 Why wouldn't the bartender say in impatience: 'Here - knock yourselves out'
@@bigmichael6156 That's a version of the famous Indian tale of 'rice on a chess board' ua-cam.com/video/71yuH365Rug/v-deo.html
"You can't really count all the natural numbers, you die at about 3 billion" - instant subscribe
"My infinity is bigger than your infinity." - Thanos
Indeed, is the Infinity Gauntlet enumerable or non-enumerable?
Don't wanna destroy 69 likes :(
Thanos is just an imaginary irrational.
"You call that big? As if." --He Who Remains (Kang)
He really needed somehelp poor guy thought 2× meant something hell he didn't even push an octave
Please, don't apologize for the length of the video. It is extremely fascinating and informative. And, infinite, of course!
In the proof at 04:22, in the first line, it must be suppose a/b = sqrt(2).
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.
@@ShaneClough what are you saying????
The supposition itself is wrong it should be a/b =sqrt2
Not a/b = 2
Shane Clough thanks
Yeah, I got really confused by the proof until I realized the first line was supposed to say sqrt(2) rather than 2.
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!
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.
This sounds more plausible for some reason
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.
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. 🙌👏👍
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.
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.
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.
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!
Yeah! Always fun to hear about Aleph null and Aleph one
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.
Fantastic video, really enjoyed it. I found all the explanation a lot simpler than other explanations I've seen before. Thanks a lot for this video Up and Atom! Cheers!
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.
Jade's enthusiasm is transcendental. For a formal proof, watch her videos!
Yay, a new Up and Atom video!
Another lovely video. I love how you show a proof by induction without getting into the guts of how to prove by induction! Brilliant.
Woah this is awesome ! I've tried a bunch of times to understand the different "sizes" of infinity and this is the first time i really get it ! Thank you so much !
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.
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.
@@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.
@@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.
@@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.
@@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.
Wow, yours presenting is beyond amazing!
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)
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...
Zeno’s paradox
And this my folks, is what you need to know to understand limits in calculus.
Hehe, and the limits of calculus.
@@NotHPotter Weierstrass function
@@stevec7819 well played
@Ian M But DC has a lot of limits xD
Hell no. That is so different. Limits are about the arbitrarily large. Whether you are talking formally or informally.
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?
the diagonalized number is not rational.
@@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?
@@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).
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.
@@DarwinsStepChildren Thank you so much man. You cleared all of my doubts
Props on the Charles Petzold quote. 'CODE' is in my top 5 most important books I've ever read.
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
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!).
Cantor was wrong, in my opinion all infinities are the same. I can prove it if you want.
This is the first time I've heard this stuff and felt like it made any kind of sense. Thank you so much!
I love your videos. You make these concepts so much easier to understand especially with your animations. The only catch I have is that the sound quality is a little muffled. The higher frequencies seem to be missing. Is there a way you can fix this?
Great video, maybe part II about continuum hypothesis and Godel incompleteness theorem?
yes i so want to cover the incompleteness theorem. just gotta wrap my head around it first
I was waiting for Godel to show up.
Which one of Gödel's three incompleteness theorems?
Reading about mathematicians you begin to wonder if going too deep into maths awakens the “Great Old Ones” and drives you insane
I love the craft of the comment
No.
only awakens one to their spirit essence... the singularity
@@reikiorgone No need for that hypothesis.
@@Raison_d-etre that's an a priori assumption... Love ya!
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 :)
So you could also say that Cantor was wrong and all infinities are the same. They are Infinite!
This is one of my favorite topics in pure math. I love your presentation. Very succinct!
All of set theory's talk of alphas, betas and omegas is Greek to me.
its literaly greek my baby
@Kat Cut It's a pun on an English idiom.
i am the alpha!
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.
It could really use some alephs.
(Just google "aleph infinity.")
4:40 Hippasus, is that really you ? 😂😂😂
Such an enjoyable concept to think about! Thanks and keep up the good work, Jade :)
Hi Mr Aly! I think you operated on me at the Austin hospital back in 2021, I had my gallbladder removed. Thank you Mr Aly! I hope that you are well. I always tell people that "Ahmed" is the best surgeon in Melbourne!! Best wishes, Emma Pasquale.
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
Actually Euclid uses the word ‘ἄλογος’ (alogos - literally ‘irrational) in The Elements.
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?
whats wrong with eating dogs?
@@Blox117 there's nothing necessarily wrong with that, but sticking forks in living dogs is cruel :-)
I hope it has nothing to do with hunger.
Well explained; I hope you create a video on the diffrence between Cardinal and Ordinal numbers, It's always pleasure to watch your videos .
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"
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.
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...
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.
@@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...
If the number of 0 is a quasi-infinite number, you just described the first rational number above 0.
@@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.
doesn't the argument starting at 7:42 hold also for just the rationals?
No, because a rational is defined by the division of two integers, which are defined by finite successors of 0.
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).
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?
@@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.
@@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
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.
Your enthusiasm is contagious! Love the videos
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.
"Some infinities are bigger than other infinities"- Fault in our stars
Jade, thanks for taking a run through this topic. It's fascinating but a lot of work. David Wallace, "Everything and More" is another resource, if you, like me, need an artifact to hold in your hands while you wrestle with these ideas. Thanks also for your link to "The Annotated Turing". I trust your recommendations and will give it a look.
Ordered!
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 UA-cam 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.
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.
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.
I didn't understand the first proof: why can't you do the same thing with fractions?
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?
You'll get an irrational number as the limit of the two sequences.
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.
@@hojasrayadas thank's for the answer, i'll have to look at the way this list is constructed to fully understand.
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
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 :(
@@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).
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. ❤❤❤
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.
"a set is a group" lol
There is a reason mathematicians use the specfic wordy definition, "a set is a collection of elements".
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).
8:41 what i drew on my math exam paper
Absolutely love the enthusiasm. I watch so many science videos and enthusiasm certainly makes them more watchable.
videos so nicely made, so evocative, keep up the good work !!!! :))
hey thank so much :)
Number theory is infinitely interesting 🙃
Ikr
hi brother ,what is difference between set theory and numbere theory?
You make brain-melting mathematics sooo sweet! :)
Great explanation really . Especially the set theory part and the proof great explanation.Superb!
This is the kind of shit Dumbledore was probably working on. Keep it up
Infinity War!
I wish you had been my math teacher in high school. I would have hung on every syllable to every word and found a reason to stay after class. I actually had an old cranky dude who hated being there and let everyone know, daily.
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.
Just found your page and it's brilliant!! Thank you
Jade's friend recieves a call:
- Hey! Guess what! My infinite is bigger than yours!! ........ Ha!!
- *Sigh*... It's 3am, Jade, go to bed...
how did you know about this conversation?
@@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
And that folks, is why the French husband is tired...
Your first proof for non-enumerability of reals seems to work equally well on rational numbers. Meaning, it's wrong
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.
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
Great content! 👍 I would like to point out that natural numbers always starts with 1. As soon as we include 0, it becomes a set of whole numbers.
"Yeah, the Greeks didn’t like irrationals"
**throw a man off a cliff because, you know... numbers**
Well played! Greeks were suppose to be the architects of reason and wisdom and yet they rage quit math and killed some poor dude.
Who is here before 100 views?
You sir are greater than infinity.
I am! After I discard the other 2,106 views as outliers, anyway.
love the videos, what program do you use to make your animations?
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?
沙比
You are infinitely adorable. I guess I am going to have watch all your videos.
This was better explained than from a teacher
She is a teacher! In fact her audience of students is probably larger than any college professor or high school teacher's.
If you haven't already done so, could you do a video on vector projection please? Your vids are considerably easier to follow than others.
I took Set Theory in college and it was the most mind warping course I ever didn't drop.
Oh yeah? Well, my transcendental number can beat up your algebraic number!
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.
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.
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
@@Lonly82Wolf What do you mean by closed set?
@@gidi5779 I mean a closed group of numbers, that you can count.
@@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.
Great video, thank you for the beautiful, amazing humour in math, definitely subscribed now! 😊❤️👍
That was trippy!
I love the your work.
Not George Cantor (pronounced like jorj). Georg Cantor, pronounced like Gay-Orc.
Gay-Org, I would say.
@@richardfarrer5616in German final stops are unvoiced. but yeah, maybe an English speaker should say gay-org
@@letheology Interesting. I hadn't really considered it. I've done a quick survey of Internet sites though and www.howtopronounce.com/german/georg/ agrees with me in spelling, although their examples are closer to your suggestion. Meanwhile forvo.com/word/georg/ has one which is definitely Gay-orc, while another is completely unvoiced with Gay-or. I can safely say that, when watching a video on infinity, I did not expect to learn something about German pronunciation.
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.
Not really, it's actually any operation mod9
I'd add 5 mod 10 (or 2 mod 4 etc.; as long as the base is at least 4, it works).
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.
This is awesome! Very well presented.
I'm totally engrossed by the philosophical side of maths. Thanks again for a great video!
thank you :)
After class, this video has cleared some doubts my doubts. Thanks a lot.😊
I've always liked concepts inside mathematics. It's really interesting hearing people talk about it. To me, it's like a pure form of logic with its own world of idiosyncrasies. It seems the more you know about how numbers are and behave, the more you realize you know very little. I loved the documentary on Andrew Wiles and how he proved Fermat's last theorem. And although I didn't understand much of the inner workings, I did understand the concepts that allowed him to prove it. Just the fact that he could even begin to relate "this with that" in such a strung out manner to even see what he was trying to go for, is just mind blowing to me. No wonder it took him years! Anyway, I digress. I just enjoy learning. Sometimes the biggest ideas come from the most humble beginnings.
totally agree with this. and I didn't know there was a documentary about andrew wiles, I'll have to check it out!
Excellent video 👌👏
To demystify the transcendentals look up Liouville numbers.
Love the quote at the end!
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?
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.)
Great video, again. Cantor was certainly a character, from his philosophical endeavors to his Baconian theories.
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.
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.
yeah it would be the same they were right :)
Creepy how Google/UA-cam knew I was reading that exact book and recommended this video! But super good recap of chapter 2 and made it easier to understand :)
Thanks Jade
Thank you for posting these videos. It is refreshing to be reminded of the absurdity of the world we live and constantly don't pay attention to. One way I try to make sense of the continuum is like this: The Number Line is an infinite line. Each number in order can be assigned a position. Between each number we can place all of the other numbers that fit in their own places. The trick though is to fit the numbers like 0.(infinity)1, 0.9(infinity)9, etc., those we can say are at a right angle in the plane of the Number Line, extending to infinity on their own number line, simply beginning where the two line are intersecting. This brings me to the irrational numbers. Now I did not think of this on my own, but was rather told by Numberphile here on youtube: they can be easily represented as a 3D line curving out of the XY Plane and into XYZ. So once you add imaginary numbers to your Plane of the Continuum, you have a 3 dimensional representation of all possible numbers that we know of so far. At least until you decide to see the universe as a point represented as 1 and add 1 to it.
Started from your recent video then came back to watch this. Really love it. Am a sophomore engineering student 🤗.
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).