Hey all! Just wanted to give a clarifying not. There is no general consensus in math as to whether zero is a natural number or not. To precisely distinguish them, one can use the phrases "non-negative integers" or "positive integers." However (!) in set theory, which is the branch of mathematics being discussed in this episode, there is a consensus: Zero is a natural number. The set theoretic constructions of the natural numbers (e.g. in the Peano Axioms) includes zero.
Gotcha. But one of the things that confused me is that she also referred to them as the "counting numbers", which is disjunct from that ISO 80000-2 zero-inclusive definition, sometimes used by logicians.
+akhil prasad sebastian - Why wouldn't it? The definition of "bijection" is independent of what size the sets involved have. There is no reason to specifically exclude infinite sets from this definition. Furthermore, the rule of bijection is the _only_ rule there is for determining if two sets are the same size. Even simple counting is merely a bijection between a subset of the natural numbers (1, 2, 3, 4, 5) and another set (the fingers on my right hand). Arbitrarily excluding infinite sets from that rule doesn't make sense. Sure, you _could_ exclude them, but you would end up with much more boring and much less useful math. As it stands, the definition of bijection works just fine for infinite sets and so does the definition of "same size".
PBS Infinite Series Could just say natural numbers include 0 and the positive integers doesn't (as 0 is a neutral not positive integer) to distinguish between the two sets...
I'm a math undergraduate student, and when I was 12 years old the subject of different infinite sets was one of the reasons I liked math so much. But I think this video is incomplete. I would have liked to see explanations for Cantor's diagonal proof and why the rational numbers are countable, and maybe an explanation of the power set and Cantor's theorem. It would be amazing if you could have that as a followup video next week. This topic is huge, and one of the most interesting ones for non-mathematicians.
Same here, that's one of the weirder parts of math and it's actually one I don't really understand. I would love to have someone explain it in a format like this.
Lemuel Ogabang I initially misread that; I thought that you meant standing up for 8 minutes (as the presenter does here) feels like standing up forever.
Oh my god. How did we not notice this before. And if you add one to 8 you get 9. Taking the square root of 9 gives you 3. The same amount of sides of a triangle! PBS Infinite Series illuminati confirmed
How can you just *assert* the real numbers are a bigger set than the natural numbers? Cantor's diagonalization proof is so simple and elegant! Why leave it out?
+DontMockMySmock You should go read the comments in Vsaurce's video about it. Non-mathematicans get very angry when you tell them one infinitiy is larger than another. It's bemusing.
+Kai Kunstmann Say what now? What does 'finitely complete' mean here? I tried to look it up but the only context I could find was topology which I know almost nothing about. What does it mean for a set to be finitely complete? And why is it pointless?
M. Scott Veach: based on another comment that Kai Kuntsmann posted on this video, my guess is that he is trying to say that the diagonal argument merely shows that the real numbers are not in one-to-one correspondence with the set of natural numbers which have finite value. In this other post, he claims that there is indeed a one-to-one correspondence between the interval (0,1) and the set of natural numbers, but you need to use natural numbers with infinitely many (nonzero) digits to do so. Of course, there is no such thing as a natural number with infinitely many (nonzero) digits, and all natural numbers have finite value, so Cantor's diagonal argument _does_ work.
The real numbers contain irrationals while the whole numbers contain rationals or fractions. Since all fractions and whole numbers can be ordered as a list, they represent a smaller infinity than the irrationals which can't be ordered as a list. They are thus a greater infinity. Moreover, you can always create a bigger infinity by creating subsets of each whole numbered set infinity. You can also create infinite subsets of those subsets. Thus you can create an infinite number of infinities.
I truly love this series. Well put together , informative, and enjoyable. I appreciate all the work you all do to put these videos out. Many thanks, MM
Nice video! but I believe you could deliver more information in a single video. When I watch pbs spacetime, every video feels to have optimal information. Spacetime uploads a video of about 11-13 mins which is nice. I do believe you could do more.
Agreed! It is also possible to add information beyond what is spoken: I like that Space Time throws up text blocks in the background (generally from a Wikipedia entry) that are readable by pausing.
It's a new channel. They're still optimizing their style and, let's say, intellectual level of the audience. She sometimes repeat stuff seconds of having saying the same thing and such, I think that's what you're feeling. Anyway, it's a great channel.
Considering this video is titled "A Hierarchy of Infinities" I'm surprised you only considered the cardinality of the integers and that of the reals. I thought you would consider whether there are cardinalities larger than that of the reals. In fact there are - an infinite number of them. Cantor proved that the power set of a given set (the power set of a set is the set of all its subsets) always has a larger cardinality than that of the original set - even if the original set is infinite. So the power set of the reals (the set of all its subsets) has a larger cardinality than that of the reals, the power set of the power set of the reals has a larger cardinality than the power set of the reals, etc. So there is an infinite number of different sizes of infinity.
@@zuccx99 Go right ahead and create a video - I am sure we all would be most interested. Post where you put your video in the comment section here. Looking forward to it.
Melinda Green I believe the continuum hypothesis implies that this is the hierarchy of infinities. But as they mentioned in the video, it would also be compatible with ZFC if there were infinities between any set and it's powerset.
I don't think that anyone has proposed any description for a set that would would have one of these in-between infinities. But for finite sets, there can be sets between a set and it's power set. For example, between 2 and 2^2=4 there is 3.
I got a B after trying like hell towards the end. I coulda woulda shoulda got an A, if I had worked a bit better from the beginning, but overall, since my professor was great, I actually really enjoyed it. : )
As a Harvard professor once said, we teach Linear Algebra like a throw-away class for the Physicists but it turns out ALL of Math is Linear Algebra--you can never know too much Linear Algebra.
It's hardly a matter of opinion. Pick a rule set. Is it true there? Since it's independent of ZFC, you are free to make the choice and then figure out what happens if it's true and what happens if it's false. Then you can pick which system is more useful for what ever it is you are trying to do at the moment.
Good thing Physics Girl mentioned this channel or I'd've not found it this early. Thanks, PG! To my point: It's a choice to accept that a bijection may define the equivalence of magnitudes of infinite sets. I for one reject this definition and hence also Cantor's diagonal argument that assumes actual infinities. I believe there's one potential infinity of one infinite magnitude that can not be actually reached even with a supertask or any thought experiment, and that an absence of a bijection is no proof that two infinities would have different magnitudes. The pigeon hole argument holds for all finite numbers, but applying it to the infinity is an error. I think that a countable infinity is an oxymoron simply because you cannot count up to a potential infinity. Infinite hierarchy of infinities is a nice playground, but alas, as already Aristotle said, "infinitum actu non datur" (there is no actual infinity). Despite mine and Aristotle's opinion I do enjoy all kinds of maths and I want to send a big thanks to PBS for creating this show!
I'd love to see some differential geometry, like Gauss' Remarkable Theorem or the Gauss-Bonnet Theorem, even if they take a few videos to build up to. I thibk it would look great animated.
What do you mean natural numbers and natural numbers are the same size? It's literally half of the natural numbers. How can that equal the same amount? Does she mean you have 100 percent of each set within each set? If that is the case, wouldn't odd numbers also be in the same hierarchy of infinities as even numbers?
The even numbers are just an example. Numbers divisible by 3 or 7, or numbers where the equation "n % 5 = 2" is true would all work. It's just about the bijection and not about the individual numbers.
It's essentially because infinity means there is no largest number. Infinity isn't really a number on its own, its more of a process that never ends. For every natural number that exists, you can pair it with an even number. If the sets ended eventually, they would be different sizes. However, since they are infinite, they must be the same size, because every element of one set can be matched to an element of the other set.
I think, infinity is *evil*. In many senses. 1) Induction proofs, as part of infinity. 2) Hilbert's Infinity Hotel. 3) Infinity in space and time. (related to Achilles paradox, and Arrow paradox) 4) Infinity in integrals, or series. 5) Banach-Tarski paradox. Everywhere it's evil. Even in Pi. Btw, fractal - is evil with evil inside.
Hoson Lam but the whole analysis (calculus) will collapse without axiom of infinity, I deem ita as necessary for now, for we don't have alternative choice now
Cauchy Riemann Agree all previous continuous calculus cannot be defined rigorously. However discrete version of the calculus with help of today computer will provide similar powerful tools to solve similar problems alternatively. If Newton has today’s computer, he may not necessary invent/define differential calculus at all.
The complex numbers are the same size as R^2 (that's two-dimensional real space, like a plane). Here's the bijection: the complex number a+bi is paired up with the point (a,b) in 2D real space. Now, how big is R^2? It's the same as one-dimensional real space, the real number line! Can you find the bijection?
It's a while since I did Maths at this level but this question looks _hard_. Is there an elegant answer? I ended up Googling and was constantly referred to the Cantor-Schröder-Bernstein Theorem. Am I missing something?
Actually not quite. The Hilbert curve is surjective, so it covers all of R^2 and does what we need. But it's not bijective because it's not 1-to-1. If you really want a bijection you can't use a continuous mapping.
The set of natural numbers is called countable or countably infinite, while the set of real numbers is called uncountable or uncountably infinite. There are other surprising results about countability and uncountability. For instance, the cartesian product of two countable sets (defined as if a is in set A, and b is in set B, then (a,b) is in the cartesian product AxB) is countable. The union of two countable sets is countable. The rational numbers are countable. Therefore the irrational numbers are uncountable. However, it turns out there are countable and uncountable partitions of the real numbers which include some infinite sets of irrational numbers; for example, the algebraic numbers include the irrational algebraic numbers, and are countable.
kj01a The surreal numbers don't form a set; they form a proper class. So, they don't have a cardinality. But, one could say that they are too big to make a set, which is tough to think about.
I think a more certain definition for describing sizes of infinities can be by their dimensions. Simply, if 'natural numbers' is a one dimensional infinity, then 'real numbers' is a two dimensional infinity because between every possible interval of the natural numbers set, there is an infinite set of numbers. Similarly, we can define a 3 dimensional infinity by describing an infinite set of numbers between every possible real number. Like the coordinate system but only more linear: Instead of describing an infinite set of numbers for every number, describe an infinite set of numbers 'between' every two numbers. In fact these can be translated to each other: for every interval you can define a regular multidimensional array by associating the infinity set to one of it's interval's endpoints
There are infinite number of rational numbers between each pair of natural numbers. But still the all the rational numbers can be paired with natural numbers - they have the same 'level of infinity'.
Eneri Giilaan are they? if so doesn't that mean any infinite series can be paired the same way? Because any infinite series should theoretically be indexable, meaning they can be paired with natural numbers
Agreed that this is a quite unintuitive - but still 1st year university math. For a relatively decent explanation search for TH-cam video: Infinity is bigger than you think - Numberphile.
very sorry for the late response. I got exams in college, I had watching that video in mind all the time but I didn't feel like it :D these exams are really exhausting, just wanted to say I will watch it sometime :) (yeah, still haven't ^^)
I love this channel. Keep up the great work! I can see you aren't - but please don't be afraid to get as technical as you like - it's awesome. Bijection - sweeeeeet.
I have a question, if we take a set of even numbers, i.e., an infinity and use bijection to pair them up with all natural numbers, such that all even numbers are paired with all natural numbers, and add an odd number, say 1, to the set of even numbers, we wouldn't be able to pair it with any natural number as all are already used. So shouldn't a set with even numbers and an additional odd number, or any number of odd numbers, be bigger than the set of natural numbers (and smaller than the set of real numbers)
Good question. Two sets are the same size if there exist _some_ bijection. So your new set - the evens and one odd - wouldn't be paired with the natural numbers using the original bijection, but we could define a new one. For example, we could take the old pairing and add one to each natural number in the pairing -- that frees up the number 1 to be paired with your extra odd number. So we have a new bijection that does work!
mina86 Yeah, I've watched the Banach-Tarski video by Vsauce. It's really cool! Also, the Hilbert's hotel is the one in the Ted-Ed's video Infinity Hotel paradox right?
Part of the 'problem' [esp for communicating with the lay person] is that 'infinity' itself isn't that well understood in the first place. The set 'goes on and on' aspect, and the separate 'counting' aspect are distinct concepts that get confounded when the set is the 'integers' that appear to match the countings. The bijection between the positive integers and the evens is between _different_ sets (and their particular orderings). Both sets 'go on and on' in a definite countable order so are of the same 'countable size'. For the rationals, the ordering isn't (for the purpose here) by linear value, rather by one of the diagonalization orders. It is that ordering which makes the set 'countable'. Having decided that one _can_ count the rationals, there is a flip to an order that doesn't appear to have the countable property (but is the same set) that is then used to show that the reals are definitely larger even though we get into the 'alternating' vs 'between' problem of reals and rationals (i.e. reals having smaller infinitesimals that the rationals ;-) If you want to further confuse the issue you get into the 1.000000... being preceded by 0.999999... for some arbitrarily small infinitessimal ! Monty-Hall had it easy.
@@farkler4785 Who is the "we" that you talk of; why does it (0.99..) keep coming up? Both are valid representations (i.e. all the base-1 repeating digit representations) that can be in used in any of the diagonalization arguments. There is a blind spot as to how 3/3 = 1 but (1/3)*3=0.99..... It's all about communicating the key steps that either suspend disbelief or 'jump the shark', or flip philosophy to explain just how certain apparent impossibilities happen (counting to infinity squared, etc.) One solution is to invoke the distinction between 'arbitrarily large' and then 'countable infinity' and how they differ as to the _conventions_ they invoke.
Who wrote the amazing intro/theme music at 10 seconds? I love it! The synthesizer has such a nice sound. Can you please attribute them in the description, I'd like to follow up and listen to more of their stuff (and hopefully a full length version of the theme).
Why can't I n++ my like if I watch the video more than once. This is a beautiful video (not just because the host is beautiful). I want to hit like more than once.
It's not that different infinities were purposely introduced (at least at first), they were fist discovered by Cantor. After that the need to formally describe what sets are arose and mathematicians simply wrote down what operations they would like to do with sets. Many different infinities came as a consequence. Most of mathematicians don't actually use huge infinities, but they can be useful, for example, set theorists use large infinities to measure how strong a mathematical theory is (in a sense what a theory can prove).
The set of all countable ordinals has a cardinality of aleph-1, therefore if the continuum hypothesis is false, it's an example of a set bigger than the naturals, but smaller than the reals. If the continuum hypothesis is true, then it's the same size as the reals.
It would be great if you could recommend a book at the end of each video that goes into greater depth over whatever topic you covered. In this case, "Uses of Infinity" by Leo Zippin is something I've heard a lot about.
Great video. I had never really thought of the boundary between regimes in such a way. Though, it is obvious in our speech as Sean Carroll describes. But yeah, I suppose we have to (i)magine or invent new ways to talk about new regimes and discuss the unique maths that reside within them.
In the bijection at 4:50: wouldn't there be two rays that intersect the semicircle but run parallel to the number line (and thus never meet it, meaning they have no pairs)? I mean this probably doesn't really affect the logic because you could always just make the semicircle into a quarter circle osth instead, but the conjecture made in the video is incorrect.
Nice job checking details! It's an open interval, so it doesn't have endpoints -- if it did, those rays would be parallel to the number line. It's also true that a closed interval is in bijection with the entire real number line, but this particular proof is for an open interval.
If a closed interval [0,1] is bijective to R, who takes the horizontal in your analogy? It seems there are exactly two elements in [0,1] that don't map to R. Without adding in ideas from the hyperreals, I can't connect the two.
You can make a bijection between [0, 1] and (0, 1). Consider function _f_ where: _f_(0) = 1/2 _f_(1) = 1/3 _f_(1/_n_) = 1/(_n_+2) for all integers _n_ greater than 1 _f_(_x_) = _x_ for all remaining real numbers in [0, 1] The function can map to any element in (0, 1) (it is *surjective*) and no two inputs will map to the same value (it is *injective*). This can be used to show the function is bijective. This also means there is a bijection from [0, 1] to the real numbers by composing the function _f_ here with the semicircle (a kind of _tan(pi/2 * (2x - 1))_) function
Is there a set that is the next infinity up from Aleph_1 (in ZFC) that can be described in any sort of intuitive way? I've been wondering this for a looooong time. Can't seem to get a straight answer. Someone once said that if you add the infinitesimals you get a set of Aleph_2, but then I think i saw a proof that even with the infinitesimals added in we are still on the Aleph_1 level.
For any cardinal kappa, the set of ordinals of cardinality strictly less than kappa, is of size kappa. In particular, omega_0 - the set of all finite ordinals - is countable, omega_1 - the set of all countable ordinals - is of size aleph_1 and omega_2 - the set of all ordinals of size at most aleph_1 - is of size aleph_2.
I believe that if you accept the continuum hypothesis, you can get a set of size Aleph_{n+1} by taking the power set of a set of size Aleph_{n}. So, a bit of a boring answer to your question would be the set of all subsets of the real numbers. A more visualizable example is the set of all functions (not necessarily continuous) from R to R. To see that this set has cardinality strictly bigger than R, you can show that it contains a copy of the power set of R. Interestingly, imposing the restriction of continuity brings the set down to cardinality equal to that of the reals.
Jordan Snyder awesome, thanks! I supposed the power set of any interval on the continuum would also be size aleph_2, right? so for example, all the intervals in -1 to 1.. or am I missing something? ohhh now that I think about it all the intervals between -1 and 1 seems like it would be less than all the possible sets of point between -1 and 1, but I don't have a good intuition of whether that set would be aleph-1 or aleph-2
This is incorrect. The continuum hypothesis says alpha_1 is the size of the reals. It does not decide the size of any of the larger cardinals. The general claim that |P(kappa)| = Kappa^+ is called the Generalised Continuum Hypothesis, and of course, it is independent of ZFC. In summary, even if CH is true then it does not follow that |P(R)| = aleph_2.
Maile, it can be shown that any open interval (a,b) has the same size of the reals. You can even find a bijection which is smooth and with smooth inverse. This, however, does not tell you where the size of the reals falls. Part of the reason that your intuition around aleph-1 and aleph-2 is that most "reasonable" sets you can write down are either finite, countable, size continuum or even bigger. For instance, it can be shown that if a subset of the reals closed, and uncountable, then it is size continuum. If it were easy to think of sets of size aleph-1 or aleph-2, we could probably decide the continuum hypothesis.
Nope, the cardinality is the same if you mean continuous or even piece-wise continuous . The set of all functions from the reals to the reals is of bigger cardinality though, 2^c, but most of the functions would just be a fog of points and not a connected curve.
I was getting ready to go off about the continuum hypothesis before I got to that part in the video, glad you guys included it! I'd be pretty impressed if you guys could tackle Galois theory
In the function y=a/x, the value soars to infinity when x approaches 0, but it happens with different speed, depending of the value of a., hence many infinities. A handy method to handle this, is to define a unit of infinity. Call it inf, and define it as such: imf=1/x when x approaches 0. Then any other infinity can be expressed as a*inf. Defined that way inf can to some extent be regarded as a number and used in calculations, even though it will not work universally due to some deficiences in the axiomatic foundations of mathematics.
So what does 2*inf stand? 1/x^2 as x go to zero? Or 1/(2x)? What does it has to do with infinity? Do you want it to be negative or positive infinity considering lim 1/x as x goes to zero is not defined because it both goes to infinity and negative infinity? Does inf*inf exist? And inf^n? And inf^2? And most important..what the helld does this has to do with infinity? Can you find a connection with the amount of things in sets?
My point is this. Infinity is not a well defined concept in standard mathematics, and hence it cannot be used to calculate with in an easy sense. What i show you is that this problem can partly be remedied by defining 1/0 as a mathematical unit called for example "inf". Then every a/x for any a will have a meaning. There will alo be a negative version defined as -1/0. which will be -inf. Inf and its negative version can then be treated as a number and calulated with, provided you take certain precautions. In standard mathematic these precautions are taken care of by saying tha 0 does not have a reiprocal. But this is only an arbitrary precautions that make mathematics very clumpsy in certain situations.But you cannot avoid certain precautions when calculating with 0 and this inf. This is because the number 0 also is in some ways logically ambiguos and the very foundation of mathematics lacks something. The precautions I am talking about can be seen in expresions like 0/0. It can be evaluated as 0/0=1/0*0=inf*0=0. But then you must avoyd evaluating it as 0/0=1.
Bijection alone doesn't convince me that the complex nums (or quaternions, etc.) are the same size as the reals. Only when I think of the Hilbert curve does it seem plausible. Because you're mapping a higher dimension object onto a lower dimension one, you see, and Hilbert showed there is a bijection between points in R^2 and points in R^1. Just having a beer here. Thnks for the beautiful video, please keep it up.
Oddly enough, when my professors covered set theory back in college (early 1980's for me), they didnt use the term bijection ... I like it. So for the bags of pennies analogy, since every penny has standard size & material (pure copper) and thus identical volume & mass, we could use any difference in either weight or displacement of water as forms of bijection in a proof that the contents of one bag was larger than another (caveat: exempting 1944 pennies made of steel instead of copper).
I was deeply confused until I realized you were talking about cardinality and not the actual size of infinity or how fast something approaches infinity. Please make a note somewhere about it but I'm looking forward to seeing followup videos!
That there is no set whose cardinality lies strictly between natural and real numbers is the continuum hypothesis; and this proposition can't be proven true or false in set theory, assuming set theory itself is consistent. In other words, the set theory axioms are insufficient to settle the question one way or other - the proposition is true in some models of the set theory and false in others.
What happens with infinity between reals and complex numbers, quaterinions, octonions and so on......also, can you make a video for ZFC with quantifiers (thanks a lot. Jan Pahl from Caracas, Venezuela)
I was disappointed that you never mentioned Cantor's diagonalization argument, because that was what got me to understand the difference between countable and uncountable infinities. Also, how it can be used to prove very intuitively that the rational numbers are countable. Also the fact that mathematicians call them countable and uncountable infinities, not "The sets of natural numbers and the set of all real numbers".
if there is cardinality in the infinities lowest (natural numbers) and highest (real numbers) then there is a cardinality in binaries the highest significant bit and the lowest significant bit? However in order to know highest and lowest bit it is not infinite any more ?? Do you know how to tie the lose ends together in retrospect of the continuum hypothesis? (and use non-standard analyses as a tool)
I've heard proper forcing axiom would allow you do prove continuum hypothesis in a way that makes sense. Maybe episode on that? tried reading about it but its complicated
“It has been said that all infinite sets are endless, but some are more endless than others.” I will try to explain where Cantor went wrong and how it IS possible to “count” the set of irrational numbers and by extension the reals. Cantor’s goal was to find a unique natural number for every real number and vice versa, it was not to find a list with a 1 to 1 correspondence between the naturals and reals (though this would have settled the issue, this would have been a sufficient condition but it was not a necessary condition). To get to his goal, Cantor could have used the Schröder-Bernstein theorem. Which says, for two sets to be equipotent, you need only show that there exists injective functions f : A → B and g : B → A then |A| = |B| It is easy to find a unique real number given a natural number, e.g. 12 → 12.0 or 12 → 12.14159… or generally n → n.xyz… where n is a natural number and xyz… is any string of digits. Or, we can find a real number in (0,1) for every natural number. e.g. 1230 → 0.0321 or 985356295141 → 0.141592653589 But, finding a unique natural number given a real number is not as obvious, yet it can be done. Cantor assumed, in his proof, that it was necessary to list all the natural numbers, in order and on one list. This assumption is erroneous. It is like a rich man going to a small country auction, if he bids all his money ($1 million) on the first item, then he will be able to buy only that one item, but if he bids in smaller increments, he will be able to buy all the items up for bids. Cantor could have paired his purported list of ‘ALL’ real numbers in (0,1) to a small subset of the natural numbers like this. 4 → 0.5123453… 44 → 0.5125674… 444 → 0.5124195… 4444 → 0.5123676… 44444 → 0.5127295… … Then when he finds, by diagonalization, that he has not listed all the real numbers we could simply say “so what, we have not listed all the natural numbers either”. But we have satisfied our condition of finding some unique natural number for each real number in the list and we still have plenty of natural numbers left over to pair with any real number that may turn up. So, Cantors proof is thus inconclusive. We can then ask Cantor to take the diagonals of the elements in his list, change the digits and make a second infinite list of real numbers that were not on his first list. And we can then pair those real numbers to some other subset of the natural numbers like this. 14 → r1 144 → r2 1444 → r3 14444 → r4 144444 → r5 … Again, we can ask Cantor to take the diagonals of the elements in his second list, change the digits and make a new infinite list of real numbers, that were not on this second list (being careful to not duplicate any numbers from his first list). And we can then pair those real numbers to yet another subset of the natural numbers like this. 24 → rr1 244 → rr2 2444 → rr3 24444 → rr4 244444 → rr5 … And, naturally, we can keep this game going forever. The interesting thing about this is, that when the prefix of our natural numbers approaches infinity, then it will become harder to find a real number, by changing the digits of the diagonal numbers, that is not on one of the infinite quantity of lists of real numbers. Also notice that most of the natural numbers like 333, 777, 123789 etc do not appear anywhere in our pairings, but that is okay, since we have enough natural numbers to pair to our sets of irrational numbers without them. All this tells us is that it is possible that the set natural numbers is bigger than that of the set of real numbers in (0,1) or that they are equal in size. Now that we have cast doubt on Cantors diagonal proof, we can show how it is possible to find a function that does pair up the natural numbers to the irrational numbers. (the rational numbers can be paired, so we will ignore them for now) We will generate a set of random irrational numbers between 0 and 1, our list might look like this. 0.5123453… 0.5125674… 0.5124195… 0.5127676… 0.5124295… 0.05127295… … Then we can imagine a deck of cards where the first card has the number 1 on it, the second card has a 2, third a 3 and so on, a stack with all the natural numbers in order. The first number on our random list is 0.5123453… so we can pair it with the fifth card in our deck. The second number on our list is 0.5125674… since we have used the 5 card, we must now use card 15. (the first two digits in reverse) The third number on our list is 0.5124195… since we have used the 5 and 15 card, we must now use card 215. (the first three digits in reverse) The fourth number on our list is 0.5127676… since we have used the 5, 15 and 215 card, we must now use card 7215. (the first four digits in reverse) The fifth number on our list is 0.5124295… since we have used the 5, 15 and 215 card, we must now use card 4215. (the first four digits in reverse) And so on. So we have our pairing. 5 → 0.5123453… 15 → 0.5125674… 215 → 0.5124195… 7215 → 0.5127676… 4215 → 0.5124295… 50 → 0.05127295… … If you ponder on it for a while, you will see that there will never be an irrational number that we can’t find a natural number to pair with it. We could sort the natural numbers and keep the irrational that it is paired with, together with it, and then look at the nth digit of the nth real number that it is associated with. For example, 1 → 0.14159265359798… 2 → 0.23606797749979… 3 → 0.3166247903554… 4 → 0.414213562373095… 5 → 0.5123453… … Diagonal = 0.13624… then change the diagonal digits to something else and we would get an irrational number that is not in our list of irrational numbers. Anti-Diagonal = 0.34333… Since the number that we create is different from every number in our list of irrational numbers, it must be different at some finite point. (it is impossible for two irrational numbers to have the first infinite number of digits in common, then to be followed by some digits that are different.) Since this diagonal defines a finite string of digits, that has not been seen on the right side of our set, then we know that same finite string does not appear on the left side either, since the left side is simply a reflection of the digits that have appeared on the right side. So we know that there exists a natural number equivalent to that string (or a substring of that string) that will be available to pair with the irrational diagonal number that introduced that finite string. Therefore, the cardinality of the two sets is the same. And the continuum hypothesis is true, there is no infinity between the naturals and the reals.
You are mistaken in a few places here. First, you say that, in your process, "it will become harder to find a real number." It won't, really. You're making it artificially harder to find a new real number by generating a large pile of lists, but one cool thing about a pile of lists is that you can make them into one list. And, in fact, by pairing up reals with naturals, you've given us a really convenient method of doing this. Just order by the sizes of the naturals. Now we have one big list, and we can easily find a new real that is on none of the subset lists. No matter how many lists you add, you will never make it any harder to find a new real. It is thus never going to be possible to get all the reals onto your list/lists. Now, let's look at the actual pairing you've generated. To break this pairing, let's stop looking at random irrationals, and start, y'know, picking them. Your pairing method should, after all, work for any order of irrationals, not just for weird orderings. The first irrational will be .1314159... This, obviously, pairs with 1 in your system. The next is .2314... And, in general, the nth element of this irrational ordering is a decimal point followed by the digits of n, followed by the digits of pi. Here's the kicker. Using only this really specific kind of irrational number, I have eaten every single natural number. You say I will never find an irrational number that I cannot pair to a natural number? I pick .114142..., or one followed by the square root of 2. This number cannot be paired with 1, for 1 was paired with .1314..., it can't be paired with 11, because 11 was paired with .11314..., it can't be paired with 114, because 114 was paired with .114314..., and so on. This irrational will never find a partner before infinite digits, and natural numbers do not have infinite digits. Your pairing method does not work. No pairing method will.
Edgar Nackenson said “ It is thus never going to be possible to get all the reals onto your list/lists.” True enough, but that is not the problem. The problem is that we have paired all the reals on infinite lists where each list has an infinite number of reals, with natural numbers and we still have lots of natural numbers left over. We have only used the natural numbers that have a 4 in the units place. As for your second argument, let me use your own words to show you why it fails. We will consider the rational numbers instead of the irrational numbers. Now, let's look at the actual pairing you've generated. To break this pairing, let's stop looking at random rationals, and start, y'know, picking them. Your pairing method should, after all, work for any order of rationals, not just for weird orderings. The first rational will be 1/1 This, obviously, pairs with 1 in your system. The next is 1/2 And, in general, the nth element of this rational ordering is a ratio of 1 divided by n. Here's the kicker. Using only this really specific kind of rational number, I have eaten every single natural number. You say I will never find a rational number that I cannot pair to a natural number? I pick 2/3. This number cannot be paired with 1, for 1 was paired with 1/1, it can't be paired with 2, because 2 was paired with 1/2, it can't be paired with 3, because 3 was paired with 1/3, and so on. This rational will never find a partner before infinite digits, and natural numbers do not have infinite digits. Your pairing method does not work. No pairing method will. Can you see how your argument makes the rational numbers uncountable? Here's the kicker. It does not matter how many strategies do not work, we are only looking for one strategy that does work, and if we find one, then the cardinalities of the two sets are equivalent.
To the first point, it is incredibly irrelevant that you're leaving a pile of natural numbers in your back pocket. You can add as many other natural numbers as you want and you'll still fail to account for all the reals. We know this for a fact, because you already essentially used all the natural numbers. After all, the natural numbers with a 4 in the units place share a cardinality with all natural numbers. To the second point, what you've said here does not function as a counterargument. The reason for this, straightforwardly, is that I haven't told you how the rationals are being paired with the natural numbers. Your mapping is fundamentally dependent on the order I present you with real numbers, which is why giving your mapping a specifically ordered pile of real numbers is a valid argument. The pairing between naturals and rationals does not function in this way. Each rational is statically paired with a specific rational in a predetermined method, such that, if you name a random rational number, the system does not default to picking the first natural number, or even the first natural number of a given form, but rather a specific natural number determined by the system. It's a bit easier to see this with a more straightforward pairing, like odds to evens. The mapping here could easily be n -> n+1. So, if you name even numbers at random, say 8,614, I don't have to blindly grab at the first odd with a few properties. I can specifically tell you that this number pairs up with 8,613. You are correct that finding individual strategies that fail to work is not sufficient, and that providing one counterexample would be more than sufficient. Proofs by counterexample are a beautiful thing, generating math truth easily and clearly. However, first, you have not provided a proof by counterexample. Your mapping doesn't work. This in itself doesn't prove that no mapping will work, but, second, something that does prove that no mapping will work is Cantor's diagonal argument. Simply take your mapping, reorder the natural numbers (preserving the mapping), and apply Cantor's diagonal argument, and you will have, again, missed 100% of the real numbers.
olavisjo Just checking, was there supposed to be a response here? Cause I got notified regarding one, and I dunno if it was deleted or is just weirdly invisible.
Can we segregate infinities of the "same" size by how fast the elements in the sets approach infinity? Meaning use "size" and "rate" to define different infinities?
No that concept doesn't apply to arbitrary sets. Not every set can be measured so the concept of "rate" doesn't apply. Some sets can't even be ordered so you can't even put them in a sequence that makes sense.
Excellent! Explicit coverage of bijection (correspondence) is often missed in other discussions of the hierarchy of infinities, though it is always implicit. Vocabulary Wish-list: 1) Use "Aleph notation" when introducing the Tower of Infinities. 2) Bijection is "pairing", but is an instance of the more general topic of "mapping". 3) Integer/Whole numbers are "countable" (in principle) where Reals are not. In general, deep understanding of a concept is less important that knowing the name of the concept (and its context), simply because we can Google for a name, but would have a tough time searching for a concept without knowing the related names and vocabulary. In other words, a concept without metadata (name and context) is isolated from associated concepts. Math is already extremely abstract: Unlike, say, physics, where a specific physical phenomenon or famous experiment can be used as a touchstone. Please use each video as an opportunity to not only examine a specific concept, but also to introduce all the relevant terminology that would make Google more useful to viewers desiring to learn more. Hmmm. There may be a future Infinite Series video here: How to use Google to learn more about a math concept.
TBH, I'm glad she avoided some of the lingo around sets, I think it can be a bit distracting when approaching this topic for the first time. This stuff can be pretty mindblowing to the uninitiated, and I think it's prudent to break their brains slowly. :) I definitely agree with #2 though, it would be nice to have had a more thorough definition of 'bijection' here, emphasizing it's not just being able to match one way.
I've a question. we know that the distance between two physical points on earth namely x0 and x1 is real infinite. And we also know that the difference between two point of time namely t0 and t1 is real infinite too. And we know that we can travel from x0 to x1 in (t1-t0) time with a constant speed. Is that means division of two real infinites may be a natural number or we are not actually moving and living in inception world (!)?
Thanks gurl. I learned a lot of math (8 semesters just from calculus) and never could grasp the continuum hypothesis. But now I think I do. My only question is, is there an alef omega infinity? An infinity which as infinitely many infinities smaller than it?
They are the same size. These "tricks" get much trickier and technical when it comes to more interesting bijections. Simply put, it is possible to fill a space of any dimension with a single curve (line which can turn) which is naturally one-dimensional.
Space-filling curves are overkill. For every real number you can take just the even numbered digits to make a real part and the odd numbered digits to make an imaginary part.
Martin Epstein That doesn't work though. There is this catch that real numbers are notexactly isomorptic to infinite decimal expressions. Take following real numbers and look where your function maps them into complex plane: 0.09090909... -> 0 + 0.9999i 0.10000000... -> 0+1i While the real numbers on the left are distinct, the complex ones on the right are identical, so it is not a bijection.
Jakub Pekárek Sure, but the same is true of space-filling curves. source: math.stackexchange.com/questions/43096/is-it-true-that-a-space-filling-curve-cannot-be-injective-everywhere I was really just going for a surjective map. By the Schroder-Bernstein theorem, if you can get a surjective map and injective map separately then you've proven the existence of a bijective one (I only learned that just now). Source: en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem
Martin Epstein What you give here talks about continuous mappings, which is different object alltogether, we are considering set to set (no structure) map. The Cantor-Bernstein theorem would work for a different setting. One way is simple as R is contained in C. The other way could be made through inverse of what you described, but it's trickier. You'd basically need to establish that the inverse construction creates distinct infinite decimal expression. Then you would need to prove that distinct obtained expressions correspond to distinct real numbers, which would be a real pain as it is not true universally. And then you could invoke the theorem. However, if you want to invoke theorems, I'd suggest saying that C = RxR while for infinities X and Y it holds |X|*|Y| = max{X,Y} (in this case even withou AC). Thus simply |C| = max{|R|} = |R|. Also, proving |X|*|Y| = max{X,Y} is simple, certainly simpler then doing stuff described above. Nevertheless the original question was geometrically oriented, thus I used curve.
Math tends not to be concerned so much with application as opposed to just finding stuff out. Oftentimes results like this are fairly 'useless' in the most obvious sense, their value is not in the result, but in the things used to prove that result. She didn't go over the proof directly in the video, but part of the proof that the Reals are larger than the Naturals is an *incredibly* useful tool for proving things called "Cantor's Diagonal Argument." This argument is a clever way of constructing new items in a set from old, such that you can show no possible bijection can exist. This is normally a pretty hard thing to do. This technique of 'diagonalization' shows up in a lot of places from Analysis to Topology and elsewhere. In addition, the concept of 'pairing as counting' (bijection) is a useful tool for another important approach to mathematics, reduction. Bijection as a tool can help us establish which sets are related, and translate problems from one set to another by means of that bijection. Often, it's valuable to do that because what might be difficult to solve in on domain (i.e., in one Set), is very easy to solve in another. Even if it's not easy enough in that first set, we can perhaps more easily solve it in another set we know bijects on to it, and so on. It lets us take a path through various mathematical environments to eventually arrive at something that makes our problem soluble. Hope that helps.
jfredett It definitely does and I appreciate the in-depth response. Hopefully I'll learn more about some of that in the following years so that the concepts as a whole make more sense. It seems like something I could really get in to. Thanks again.
+Majin Krocket I'm glad it helped. Math is really a very different sort of field, it's equal parts abstract nonsense and concrete, useful tools for understanding the world and how it works. If you're self-studying (heck, even if you're not) there's a lot of good material around the internet for learning about the various fields in detail, +3blue1brown is a *phenomenal* channel for survey level stuff about math, highly recommend checking him out.
jfredett I just recently found 3blue1brown about a month ago but have watched channels like numberphile, vsauce, vihart, etc. for a long time. I mentioned learning it over the next few years because I'm going to do a math minor for my CPE degree. I'm really excited for some of the more in-depth classes.
Like already said, bijections are incredibly important everywhere in maths. In maths you are always interested wether you can see two objects you defined as the "same". You do that by using bijections which have special properties depending on the objects you are looking at. So bijections are in a way the weakest type of "equality" of objects, the equality of sets concerning how many elements there are. The most usefull application of knowing which sets have the same size is probably saying that these to objects can't have a bijection of any type between them, so especially the ones you are looking for.
Yes, any bounded interval of reals can be mapped one-to-one with the set of all positive real numbers, or even with the set of all real numbers. But you may have been asking a different question. Can an interval of reals be mapped one-to-one with natural numbers? And to that the answer is no; given any function from natural numbers to real numbers (or to an interval of reals), there is some element of the latter set which the function doesn't cover (and, by extension of the same proof, the set of uncovered numbers can be mapped one-to-one with real numbers).
Hey all! Just wanted to give a clarifying not. There is no general consensus in math as to whether zero is a natural number or not. To precisely distinguish them, one can use the phrases "non-negative integers" or "positive integers." However (!) in set theory, which is the branch of mathematics being discussed in this episode, there is a consensus: Zero is a natural number. The set theoretic constructions of the natural numbers (e.g. in the Peano Axioms) includes zero.
logicians count 0 as natural number and analysts don't, it makes sense if you think about it
Gotcha. But one of the things that confused me is that she also referred to them as the "counting numbers", which is disjunct from that ISO 80000-2 zero-inclusive definition, sometimes used by logicians.
hi just whated to ask you something
what if the so called rule of bijection
dont apply to infinite sets..
+akhil prasad sebastian - Why wouldn't it? The definition of "bijection" is independent of what size the sets involved have. There is no reason to specifically exclude infinite sets from this definition.
Furthermore, the rule of bijection is the _only_ rule there is for determining if two sets are the same size. Even simple counting is merely a bijection between a subset of the natural numbers (1, 2, 3, 4, 5) and another set (the fingers on my right hand). Arbitrarily excluding infinite sets from that rule doesn't make sense.
Sure, you _could_ exclude them, but you would end up with much more boring and much less useful math. As it stands, the definition of bijection works just fine for infinite sets and so does the definition of "same size".
PBS Infinite Series
Could just say natural numbers include 0 and the positive integers doesn't (as 0 is a neutral not positive integer) to distinguish between the two sets...
I'm a math undergraduate student, and when I was 12 years old the subject of different infinite sets was one of the reasons I liked math so much. But I think this video is incomplete. I would have liked to see explanations for Cantor's diagonal proof and why the rational numbers are countable, and maybe an explanation of the power set and Cantor's theorem. It would be amazing if you could have that as a followup video next week. This topic is huge, and one of the most interesting ones for non-mathematicians.
There are vast of topics. I guess she could talk whole time only about infinity :D.
She says there's a link in the description to a rigorous proof, presumably that's Cantor's.
Totally! The whole series could be about infinity! Y'all have great suggestions for thing you'd like to see in follow-up videos
+ imaginary and complex planes?
He said next week. :)
"In english, this means that..."
Haha, I've used this phrase too when explaining concepts after a formal definition!
lol
I just realized that the definition of cardinality actually relates really well to the Pigeonhole Principle.
Man I hope you go into the incompleteness theorem more!
^ This. I'd love to see a Infinite Series video on that, with cool animation and shit.
DekuStickGamer Ya, it is one of those areas I can't hear enough about because it is just so....odd.
Same here, that's one of the weirder parts of math and it's actually one I don't really understand. I would love to have someone explain it in a format like this.
I'd like it, too!
***** Gödel escher and bach would be a good place to start.
Avg. length of each infinite series vid is about 8 min. 8 is like infinity standing up.
Standupmaths?
Lemuel Ogabang I initially misread that; I thought that you meant standing up for 8 minutes (as the presenter does here) feels like standing up forever.
Lemuel Ogabang This is so nerdy and great that I have to reply, but I don't know what to reply with except this.
What about 24.
Oh my god. How did we not notice this before. And if you add one to 8 you get 9. Taking the square root of 9 gives you 3. The same amount of sides of a triangle! PBS Infinite Series illuminati confirmed
How can you just *assert* the real numbers are a bigger set than the natural numbers? Cantor's diagonalization proof is so simple and elegant! Why leave it out?
+DontMockMySmock You should go read the comments in Vsaurce's video about it. Non-mathematicans get very angry when you tell them one infinitiy is larger than another. It's bemusing.
Cantor's diagonalization proof only proves that the infinite list of real numbers is not finitely complete, which is kind of pointless.
+Kai Kunstmann Say what now? What does 'finitely complete' mean here? I tried to look it up but the only context I could find was topology which I know almost nothing about. What does it mean for a set to be finitely complete? And why is it pointless?
M. Scott Veach: based on another comment that Kai Kuntsmann posted on this video, my guess is that he is trying to say that the diagonal argument merely shows that the real numbers are not in one-to-one correspondence with the set of natural numbers which have finite value. In this other post, he claims that there is indeed a one-to-one correspondence between the interval (0,1) and the set of natural numbers, but you need to use natural numbers with infinitely many (nonzero) digits to do so.
Of course, there is no such thing as a natural number with infinitely many (nonzero) digits, and all natural numbers have finite value, so Cantor's diagonal argument _does_ work.
+Muffins aha.. in other words, i can safely ignore.
"There're infinitely many sizes of infinities." Are there countably or uncountably many different infinities?
Uncountably many
Equivalent to continuum hypothesis tho
If you got enough time, then yes, countable. 😅
fun fact: both
Much much more than either
I didn't understand this when vsauce or vihart explained it, but I get it now! Thanks!
V’s always make things confusing
+Jan Coker
Yeah, he should just stop doing what he does. He shouldn't confuse people.
@@romanski5811 no he shouldn't, there are plenty of people who do get informed. There are other peopme you can go to but he doesnt need to stop
The real numbers contain irrationals while the whole numbers contain rationals or fractions. Since all fractions and whole numbers can be ordered as a list, they represent a smaller infinity than the irrationals which can't be ordered as a list. They are thus a greater infinity. Moreover, you can always create a bigger infinity by creating subsets of each whole numbered set infinity. You can also create infinite subsets of those subsets. Thus you can create an infinite number of infinities.
Love this channel!
Keep up the good work!
I truly love this series. Well put together , informative, and enjoyable. I appreciate all the work you all do to put these videos out. Many thanks, MM
Nice video! but I believe you could deliver more information in a single video. When I watch pbs spacetime, every video feels to have optimal information. Spacetime uploads a video of about 11-13 mins which is nice. I do believe you could do more.
Agreed! It is also possible to add information beyond what is spoken: I like that Space Time throws up text blocks in the background (generally from a Wikipedia entry) that are readable by pausing.
It's a new channel. They're still optimizing their style and, let's say, intellectual level of the audience. She sometimes repeat stuff seconds of having saying the same thing and such, I think that's what you're feeling.
Anyway, it's a great channel.
I had to watch the video many times over to truly appreciate the writing. Awesome stuff !
Are the complex numbers larger than the real number or the same size?
They are the same size
Are there larger infinities?
The class of all sizes of infinities is so big, it doesn't even fit in a set!
What's another example other than the two given in this show?
Such a good question! There are. So many of them! They just keep going up and up and up... which would be a great future topic.
Ever since finding this concept in like, first year math, I found it fascinating!!
Vsauese's video "count past infinity" is an other awesome video of the subject.
Bálint Áts but he actually to much talking rather give a argument.
Bálint Áts the vsauce video is better than this one.
Vsauce videos are non educated and just for 'show' like common drama.
Considering this video is titled "A Hierarchy of Infinities" I'm surprised you only considered the cardinality of the integers and that of the reals. I thought you would consider whether there are cardinalities larger than that of the reals. In fact there are - an infinite number of them. Cantor proved that the power set of a given set (the power set of a set is the set of all its subsets) always has a larger cardinality than that of the original set - even if the original set is infinite. So the power set of the reals (the set of all its subsets) has a larger cardinality than that of the reals, the power set of the power set of the reals has a larger cardinality than the power set of the reals, etc. So there is an infinite number of different sizes of infinity.
No mention of aleph?
Sad.
@@zuccx99 Go right ahead and create a video - I am sure we all would be most interested. Post where you put your video in the comment section here. Looking forward to it.
No, it too Borges.
ooops (sic) it's
0:57 It took literally less than 1 minute to blow my mind this time!
I thought there was a natural ordering of the infinities which is that each higher infinity is the power set of the previous one.
Melinda Green I believe the continuum hypothesis implies that this is the hierarchy of infinities. But as they mentioned in the video, it would also be compatible with ZFC if there were infinities between any set and it's powerset.
Shawn Ligocki
What would such a fractional infinity even look like?
I don't think that anyone has proposed any description for a set that would would have one of these in-between infinities. But for finite sets, there can be sets between a set and it's power set. For example, between 2 and 2^2=4 there is 3.
Shawn Ligocki
I can't get my head around the idea. It sounds like someone saying that maybe round squares exist but nobody has found one yet.
No, it does not follow even if you assume continuum hypothesis. Your statement is the so called generalized continuum hypothesis.
Just finished by linear algebra final, now I finally get to relax while learning math : ))))
How did it go? I loved linear algebra!
I got a B after trying like hell towards the end. I coulda woulda shoulda got an A, if I had worked a bit better from the beginning, but overall, since my professor was great, I actually really enjoyed it. : )
As a Harvard professor once said, we teach Linear Algebra like a throw-away class for the Physicists but it turns out ALL of Math is Linear Algebra--you can never know too much Linear Algebra.
That makes a lot of sense, haha.
It's hardly a matter of opinion. Pick a rule set. Is it true there?
Since it's independent of ZFC, you are free to make the choice and then figure out what happens if it's true and what happens if it's false. Then you can pick which system is more useful for what ever it is you are trying to do at the moment.
not really
Many important mathematical results are related to the specific ZFC model
Good thing Physics Girl mentioned this channel or I'd've not found it this early. Thanks, PG! To my point: It's a choice to accept that a bijection may define the equivalence of magnitudes of infinite sets. I for one reject this definition and hence also Cantor's diagonal argument that assumes actual infinities. I believe there's one potential infinity of one infinite magnitude that can not be actually reached even with a supertask or any thought experiment, and that an absence of a bijection is no proof that two infinities would have different magnitudes. The pigeon hole argument holds for all finite numbers, but applying it to the infinity is an error. I think that a countable infinity is an oxymoron simply because you cannot count up to a potential infinity. Infinite hierarchy of infinities is a nice playground, but alas, as already Aristotle said, "infinitum actu non datur" (there is no actual infinity). Despite mine and Aristotle's opinion I do enjoy all kinds of maths and I want to send a big thanks to PBS for creating this show!
I'd love to see some differential geometry, like Gauss' Remarkable Theorem or the Gauss-Bonnet Theorem, even if they take a few videos to build up to. I thibk it would look great animated.
Great suggestions! Thanks
What a fantastic series! Thanks for helping me explain to my family what I love and want to spend my life doing!
Is there any proof that the natural number infinity is the smallest? Or is that just assumed?
Thank you Kelsey. Amazing show.
What do you mean natural numbers and natural numbers are the same size? It's literally half of the natural numbers. How can that equal the same amount? Does she mean you have 100 percent of each set within each set? If that is the case, wouldn't odd numbers also be in the same hierarchy of infinities as even numbers?
The even numbers are just an example. Numbers divisible by 3 or 7, or numbers where the equation "n % 5 = 2" is true would all work. It's just about the bijection and not about the individual numbers.
It's essentially because infinity means there is no largest number. Infinity isn't really a number on its own, its more of a process that never ends. For every natural number that exists, you can pair it with an even number. If the sets ended eventually, they would be different sizes. However, since they are infinite, they must be the same size, because every element of one set can be matched to an element of the other set.
Ian Krasnow What sets definitely end?
Finite sets
Ian Krasnow some examples?
The explanations are really clear. I actually understand the Video's context!
I think, infinity is *evil*.
In many senses.
1) Induction proofs, as part of infinity.
2) Hilbert's Infinity Hotel.
3) Infinity in space and time. (related to Achilles paradox, and Arrow paradox)
4) Infinity in integrals, or series.
5) Banach-Tarski paradox.
Everywhere it's evil. Even in Pi.
Btw, fractal - is evil with evil inside.
Without evil, you can't even calculate the area of a circle. we need the evilness
Yes. We don’t need this concept in mathematics with computer today. All are finite and human being is keep advancing ahead!
Cauchy Riemann In fact, the area we got was always not right but just approximation only.
Hoson Lam but the whole analysis (calculus) will collapse without axiom of infinity, I deem ita as necessary for now, for we don't have alternative choice now
Cauchy Riemann Agree all previous continuous calculus cannot be defined rigorously. However discrete version of the calculus with help of today computer will provide similar powerful tools to solve similar problems alternatively. If Newton has today’s computer, he may not necessary invent/define differential calculus at all.
Beyond Infinity!
What about complex numbers?
The complex numbers are the same size as R^2 (that's two-dimensional real space, like a plane). Here's the bijection: the complex number a+bi is paired up with the point (a,b) in 2D real space. Now, how big is R^2? It's the same as one-dimensional real space, the real number line! Can you find the bijection?
I'm gonna think about it a little bit. By the way, the videos are awesome and the content is extremely interesting, congratulations.
It's a while since I did Maths at this level but this question looks _hard_. Is there an elegant answer? I ended up Googling and was constantly referred to the Cantor-Schröder-Bernstein Theorem. Am I missing something?
Luis Vasconcellos The Hilbert curve does exactly that. A bijection betwen 1D and 2D space.
Actually not quite. The Hilbert curve is surjective, so it covers all of R^2 and does what we need. But it's not bijective because it's not 1-to-1. If you really want a bijection you can't use a continuous mapping.
Glad she covered the continuum hupothesis! it's fascinating how it could true or false, but it doesn't matter!
that gap tho :o
The set of natural numbers is called countable or countably infinite, while the set of real numbers is called uncountable or uncountably infinite. There are other surprising results about countability and uncountability. For instance, the cartesian product of two countable sets (defined as if a is in set A, and b is in set B, then (a,b) is in the cartesian product AxB) is countable. The union of two countable sets is countable. The rational numbers are countable. Therefore the irrational numbers are uncountable. However, it turns out there are countable and uncountable partitions of the real numbers which include some infinite sets of irrational numbers; for example, the algebraic numbers include the irrational algebraic numbers, and are countable.
This video is infinite confusion
The intro music to this channel is kind of badass
She didn't even get into surreal numbers, because she didn't want to get brain matter all over your keyboard.
kj01a The surreal numbers don't form a set; they form a proper class. So, they don't have a cardinality. But, one could say that they are too big to make a set, which is tough to think about.
Mind Blown. Totally loved this.
10 seconds in and my brain already exploded
This is an excellent explanation of cardinality. Amazing job!
One quick note, the set of natural numbers, denoted N, actually doesn't include 0. The next set, the integers, is when 0 is added.
I think a more certain definition for describing sizes of infinities can be by their dimensions. Simply, if 'natural numbers' is a one dimensional infinity, then 'real numbers' is a two dimensional infinity because between every possible interval of the natural numbers set, there is an infinite set of numbers. Similarly, we can define a 3 dimensional infinity by describing an infinite set of numbers between every possible real number. Like the coordinate system but only more linear: Instead of describing an infinite set of numbers for every number, describe an infinite set of numbers 'between' every two numbers. In fact these can be translated to each other: for every interval you can define a regular multidimensional array by associating the infinity set to one of it's interval's endpoints
There are infinite number of rational numbers between each pair of natural numbers. But still the all the rational numbers can be paired with natural numbers - they have the same 'level of infinity'.
Eneri Giilaan are they? if so doesn't that mean any infinite series can be paired the same way? Because any infinite series should theoretically be indexable, meaning they can be paired with natural numbers
Agreed that this is a quite unintuitive - but still 1st year university math.
For a relatively decent explanation search for TH-cam video: Infinity is bigger than you think - Numberphile.
very sorry for the late response. I got exams in college, I had watching that video in mind all the time but I didn't feel like it :D these exams are really exhausting, just wanted to say I will watch it sometime :) (yeah, still haven't ^^)
The background music is beautiful.
0:07 ... Well what is the size of the set of all infinities? What type of infinity do we assign to the "number" of infinities?
It's a properclass, a collection that is too large to be a set. So a cardinality cannot be assigned to it (this is the elementary explanation).
I love this video. the fun stuff we did in my proof classes long ago.
I LOVE THIS CHANNEL ❤❤❤❤❤❤❤
Just subbed due to this amazing video and the clarity in which it was presented! Great work
I love this channel. Keep up the great work! I can see you aren't - but please don't be afraid to get as technical as you like - it's awesome. Bijection - sweeeeeet.
I love your hair!
"All Brontosaures are thin at one end, much much thicker in the middle, and then thin again at the far end." - Anne Elk
I have a question, if we take a set of even numbers, i.e., an infinity and use bijection to pair them up with all natural numbers, such that all even numbers are paired with all natural numbers, and add an odd number, say 1, to the set of even numbers, we wouldn't be able to pair it with any natural number as all are already used. So shouldn't a set with even numbers and an additional odd number, or any number of odd numbers, be bigger than the set of natural numbers (and smaller than the set of real numbers)
Good question. Two sets are the same size if there exist _some_ bijection. So your new set - the evens and one odd - wouldn't be paired with the natural numbers using the original bijection, but we could define a new one. For example, we could take the old pairing and add one to each natural number in the pairing -- that frees up the number 1 to be paired with your extra odd number. So we have a new bijection that does work!
Oh yeah, that makes sense. Thanks a lot!
Check out Hilbert’s hotel. Or if you want something more hard-core, Vsauce’s video on the Banach-Tarski paradox.
mina86 Yeah, I've watched the Banach-Tarski video by Vsauce. It's really cool! Also, the Hilbert's hotel is the one in the Ted-Ed's video Infinity Hotel paradox right?
Yep.
Part of the 'problem' [esp for communicating with the lay person] is that 'infinity' itself isn't that well understood in the first place. The set 'goes on and on' aspect, and the separate 'counting' aspect are distinct concepts that get confounded when the set is the 'integers' that appear to match the countings.
The bijection between the positive integers and the evens is between _different_ sets (and their particular orderings). Both sets 'go on and on' in a definite countable order so are of the same 'countable size'.
For the rationals, the ordering isn't (for the purpose here) by linear value, rather by one of the diagonalization orders. It is that ordering which makes the set 'countable'.
Having decided that one _can_ count the rationals, there is a flip to an order that doesn't appear to have the countable property (but is the same set) that is then used to show that the reals are definitely larger even though we get into the 'alternating' vs 'between' problem of reals and rationals (i.e. reals having smaller infinitesimals that the rationals ;-)
If you want to further confuse the issue you get into the 1.000000... being preceded by 0.999999... for some arbitrarily small infinitessimal ! Monty-Hall had it easy.
Well we know that 1.0 is EXACTLY equal to 0.99999..., they are the same number
@@farkler4785 Who is the "we" that you talk of; why does it (0.99..) keep coming up? Both are valid representations (i.e. all the base-1 repeating digit representations) that can be in used in any of the diagonalization arguments. There is a blind spot as to how 3/3 = 1 but (1/3)*3=0.99.....
It's all about communicating the key steps that either suspend disbelief or 'jump the shark', or flip philosophy to explain just how certain apparent impossibilities happen (counting to infinity squared, etc.)
One solution is to invoke the distinction between 'arbitrarily large' and then 'countable infinity' and how they differ as to the _conventions_ they invoke.
@@philipoakley5498 I’m not following how any of this relates to cantors diagonalization theory. And the “we” is mathematicians as a whole
Who wrote the amazing intro/theme music at 10 seconds? I love it! The synthesizer has such a nice sound. Can you please attribute them in the description, I'd like to follow up and listen to more of their stuff (and hopefully a full length version of the theme).
Why can't I n++ my like if I watch the video more than once. This is a beautiful video (not just because the host is beautiful). I want to hit like more than once.
Did the introduction of so many infinities into mathematics give rise to any useful conclusions?
Asking the real questions ^
It's not that different infinities were purposely introduced (at least at first), they were fist discovered by Cantor. After that the need to formally describe what sets are arose and mathematicians simply wrote down what operations they would like to do with sets. Many different infinities came as a consequence. Most of mathematicians don't actually use huge infinities, but they can be useful, for example, set theorists use large infinities to measure how strong a mathematical theory is (in a sense what a theory can prove).
The set of all countable ordinals has a cardinality of aleph-1, therefore if the continuum hypothesis is false, it's an example of a set bigger than the naturals, but smaller than the reals. If the continuum hypothesis is true, then it's the same size as the reals.
awesome, great channel with a lot of potential in sense of added value (looking at the CC, PBS, CGP grey landscape).
Simliar to hierarchy of infinities, there is a concept called zeroness of a zero.
Excellent, I want more!
It would be great if you could recommend a book at the end of each video that goes into greater depth over whatever topic you covered. In this case, "Uses of Infinity" by Leo Zippin is something I've heard a lot about.
Great video. I had never really thought of the boundary between regimes in such a way. Though, it is obvious in our speech as Sean Carroll describes. But yeah, I suppose we have to (i)magine or invent new ways to talk about new regimes and discuss the unique maths that reside within them.
In the bijection at 4:50: wouldn't there be two rays that intersect the semicircle but run parallel to the number line (and thus never meet it, meaning they have no pairs)? I mean this probably doesn't really affect the logic because you could always just make the semicircle into a quarter circle osth instead, but the conjecture made in the video is incorrect.
Nice job checking details! It's an open interval, so it doesn't have endpoints -- if it did, those rays would be parallel to the number line. It's also true that a closed interval is in bijection with the entire real number line, but this particular proof is for an open interval.
If a closed interval [0,1] is bijective to R, who takes the horizontal in your analogy? It seems there are exactly two elements in [0,1] that don't map to R.
Without adding in ideas from the hyperreals, I can't connect the two.
You can make a bijection between [0, 1] and (0, 1).
Consider function _f_ where:
_f_(0) = 1/2
_f_(1) = 1/3
_f_(1/_n_) = 1/(_n_+2) for all integers _n_ greater than 1
_f_(_x_) = _x_ for all remaining real numbers in [0, 1]
The function can map to any element in (0, 1) (it is *surjective*) and no two inputs will map to the same value (it is *injective*). This can be used to show the function is bijective. This also means there is a bijection from [0, 1] to the real numbers by composing the function _f_ here with the semicircle (a kind of _tan(pi/2 * (2x - 1))_) function
Clever. Thank you.
Is there a set that is the next infinity up from Aleph_1 (in ZFC) that can be described in any sort of intuitive way? I've been wondering this for a looooong time. Can't seem to get a straight answer. Someone once said that if you add the infinitesimals you get a set of Aleph_2, but then I think i saw a proof that even with the infinitesimals added in we are still on the Aleph_1 level.
For any cardinal kappa, the set of ordinals of cardinality strictly less than kappa, is of size kappa. In particular, omega_0 - the set of all finite ordinals - is countable, omega_1 - the set of all countable ordinals - is of size aleph_1 and omega_2 - the set of all ordinals of size at most aleph_1 - is of size aleph_2.
I believe that if you accept the continuum hypothesis, you can get a set of size Aleph_{n+1} by taking the power set of a set of size Aleph_{n}. So, a bit of a boring answer to your question would be the set of all subsets of the real numbers.
A more visualizable example is the set of all functions (not necessarily continuous) from R to R. To see that this set has cardinality strictly bigger than R, you can show that it contains a copy of the power set of R.
Interestingly, imposing the restriction of continuity brings the set down to cardinality equal to that of the reals.
Jordan Snyder awesome, thanks! I supposed the power set of any interval on the continuum would also be size aleph_2, right? so for example, all the intervals in -1 to 1.. or am I missing something? ohhh now that I think about it all the intervals between -1 and 1 seems like it would be less than all the possible sets of point between -1 and 1, but I don't have a good intuition of whether that set would be aleph-1 or aleph-2
This is incorrect. The continuum hypothesis says alpha_1 is the size of the reals. It does not decide the size of any of the larger cardinals. The general claim that |P(kappa)| = Kappa^+ is called the Generalised Continuum Hypothesis, and of course, it is independent of ZFC.
In summary, even if CH is true then it does not follow that |P(R)| = aleph_2.
Maile, it can be shown that any open interval (a,b) has the same size of the reals. You can even find a bijection which is smooth and with smooth inverse. This, however, does not tell you where the size of the reals falls.
Part of the reason that your intuition around aleph-1 and aleph-2 is that most "reasonable" sets you can write down are either finite, countable, size continuum or even bigger. For instance, it can be shown that if a subset of the reals closed, and uncountable, then it is size continuum. If it were easy to think of sets of size aleph-1 or aleph-2, we could probably decide the continuum hypothesis.
I was about 15 when I encountered Aleph Null, Beth...etc... Mind blowing stuff (in a book in Leeds University library)
its kinda sad they just end this series , it was a good one
Amazing Channel!
I have thought about this for a while, it seems like you paired the interval between 0 and 1 with every integer and number between
The rational numbers and the natural numbers are both size Aleph Null right? I believe I saw a diagonal-style pairing back in a math class.
I understand that we don't know whether the continuum hypothesis can be proven (or disproven) under other set of rules (other than ZFC). Am I right?
On the curved interval between 0 and 1 would infinity have a ray parallel to to the line or an infinitely small angle or are they the same?
It was described as an open interval. So the endpoints are excluded.
So are there more curves then there are points on a straight line?
Nope, the cardinality is the same if you mean continuous or even piece-wise continuous . The set of all functions from the reals to the reals is of bigger cardinality though, 2^c, but most of the functions would just be a fog of points and not a connected curve.
Thanks, I'm really going to have to think about your answer.
I was getting ready to go off about the continuum hypothesis before I got to that part in the video, glad you guys included it! I'd be pretty impressed if you guys could tackle Galois theory
lol, I had the exactly same reaction
In the function y=a/x, the value soars to infinity when x approaches 0, but it happens with different speed, depending of the value of a., hence many infinities. A handy method to handle this, is to define a unit of infinity. Call it inf, and define it as such: imf=1/x when x approaches 0. Then any other infinity can be expressed as a*inf. Defined that way inf can to some extent be regarded as a number and used in calculations, even though it will not work universally due to some deficiences in the axiomatic foundations of mathematics.
No. Just no
Yes, just yes. But you must handle it with care because mathematic is cracky in its basic foundings, so you cannot use it without precautions.
So what does 2*inf stand? 1/x^2 as x go to zero? Or 1/(2x)?
What does it has to do with infinity? Do you want it to be negative or positive infinity considering lim 1/x as x goes to zero is not defined because it both goes to infinity and negative infinity?
Does inf*inf exist? And inf^n? And inf^2?
And most important..what the helld does this has to do with infinity? Can you find a connection with the amount of things in sets?
My point is this. Infinity is not a well defined concept in standard mathematics, and hence it cannot be used to calculate with in an easy sense. What i show you is that this problem can partly be remedied by defining 1/0 as a mathematical unit called for example "inf". Then every a/x for any a will have a meaning. There will alo be a negative version defined as -1/0. which will be -inf. Inf and its negative version can then be treated as a number and calulated with, provided you take certain precautions. In standard mathematic these precautions are taken care of by saying tha 0 does not have a reiprocal. But this is only an arbitrary precautions that make mathematics very clumpsy in certain situations.But you cannot avoid certain precautions when calculating with 0 and this inf. This is because the number 0 also is in some ways logically ambiguos and the very foundation of mathematics lacks something.
The precautions I am talking about can be seen in expresions like 0/0. It can be evaluated as 0/0=1/0*0=inf*0=0. But then you must avoyd evaluating it as 0/0=1.
Hey Knut. Let's chat about your idea of defining 1/0. I've studied those attempts for a long time and feel close to a useful framework.
Bijection alone doesn't convince me that the complex nums (or quaternions, etc.) are
the same size as the reals. Only when I think of the Hilbert curve does
it seem plausible. Because you're mapping a higher dimension object onto a lower dimension one, you see, and Hilbert showed there is a bijection between points in R^2 and points in R^1. Just having a beer here. Thnks for the beautiful video, please keep it up.
the definition of "having the same size" is "there exists a bijection". Any other thing than that would be a completely different concept of "size"
What are some great textbooks on this subject?
Oddly enough, when my professors covered set theory back in college (early 1980's for me), they didnt use the term bijection ... I like it.
So for the bags of pennies analogy, since every penny has standard size & material (pure copper) and thus identical volume & mass, we could use any difference in either weight or displacement of water as forms of bijection in a proof that the contents of one bag was larger than another (caveat: exempting 1944 pennies made of steel instead of copper).
In the '70s they used "bijection".
I was deeply confused until I realized you were talking about cardinality and not the actual size of infinity or how fast something approaches infinity. Please make a note somewhere about it but I'm looking forward to seeing followup videos!
Where do you get that type of background music?
Can you please speak more to the other models of infinity towers and what infinities they place in-between Uncountable and Countable infinities?
That there is no set whose cardinality lies strictly between natural and real numbers is the continuum hypothesis; and this proposition can't be proven true or false in set theory, assuming set theory itself is consistent. In other words, the set theory axioms are insufficient to settle the question one way or other - the proposition is true in some models of the set theory and false in others.
would a prime number infinity be larger or smaller than real or natural numbers?
The infinity of prime numbers is the same as natural numbers
What happens with infinity between reals and complex numbers, quaterinions, octonions and so on......also, can you make a video for ZFC with quantifiers (thanks a lot. Jan Pahl from Caracas, Venezuela)
thanks a lot
I was disappointed that you never mentioned Cantor's diagonalization argument, because that was what got me to understand the difference between countable and uncountable infinities. Also, how it can be used to prove very intuitively that the rational numbers are countable. Also the fact that mathematicians call them countable and uncountable infinities, not "The sets of natural numbers and the set of all real numbers".
if there is cardinality in the infinities lowest (natural numbers) and highest (real numbers)
then there is a cardinality in binaries the highest significant bit and the lowest significant bit?
However in order to know highest and lowest bit it is not infinite any more ??
Do you know how to tie the lose ends together in retrospect of the continuum hypothesis?
(and use non-standard analyses as a tool)
I've heard proper forcing axiom would allow you do prove continuum hypothesis in a way that makes sense. Maybe episode on that? tried reading about it but its complicated
Its 3am and i don't know why im doing this to myself
But i can't stop
“It has been said that all infinite sets are endless, but some are more endless than others.”
I will try to explain where Cantor went wrong and how it IS possible to “count” the set of irrational numbers and by extension the reals.
Cantor’s goal was to find a unique natural number for every real number and vice versa, it was not to find a list with a 1 to 1 correspondence between the naturals and reals (though this would have settled the issue, this would have been a sufficient condition but it was not a necessary condition). To get to his goal, Cantor could have used the Schröder-Bernstein theorem. Which says, for two sets to be equipotent, you need only show that there exists injective functions f : A → B and g : B → A then |A| = |B|
It is easy to find a unique real number given a natural number, e.g. 12 → 12.0 or 12 → 12.14159… or generally n → n.xyz… where n is a natural number and xyz… is any string of digits. Or, we can find a real number in (0,1) for every natural number. e.g. 1230 → 0.0321 or 985356295141 → 0.141592653589
But, finding a unique natural number given a real number is not as obvious, yet it can be done.
Cantor assumed, in his proof, that it was necessary to list all the natural numbers, in order and on one list. This assumption is erroneous. It is like a rich man going to a small country auction, if he bids all his money ($1 million) on the first item, then he will be able to buy only that one item, but if he bids in smaller increments, he will be able to buy all the items up for bids.
Cantor could have paired his purported list of ‘ALL’ real numbers in (0,1) to a small subset of the natural numbers like this.
4 → 0.5123453…
44 → 0.5125674…
444 → 0.5124195…
4444 → 0.5123676…
44444 → 0.5127295…
…
Then when he finds, by diagonalization, that he has not listed all the real numbers we could simply say “so what, we have not listed all the natural numbers either”. But we have satisfied our condition of finding some unique natural number for each real number in the list and we still have plenty of natural numbers left over to pair with any real number that may turn up. So, Cantors proof is thus inconclusive.
We can then ask Cantor to take the diagonals of the elements in his list, change the digits and make a second infinite list of real numbers that were not on his first list. And we can then pair those real numbers to some other subset of the natural numbers like this.
14 → r1
144 → r2
1444 → r3
14444 → r4
144444 → r5
…
Again, we can ask Cantor to take the diagonals of the elements in his second list, change the digits and make a new infinite list of real numbers, that were not on this second list (being careful to not duplicate any numbers from his first list). And we can then pair those real numbers to yet another subset of the natural numbers like this.
24 → rr1
244 → rr2
2444 → rr3
24444 → rr4
244444 → rr5
…
And, naturally, we can keep this game going forever. The interesting thing about this is, that when the prefix of our natural numbers approaches infinity, then it will become harder to find a real number, by changing the digits of the diagonal numbers, that is not on one of the infinite quantity of lists of real numbers. Also notice that most of the natural numbers like 333, 777, 123789 etc do not appear anywhere in our pairings, but that is okay, since we have enough natural numbers to pair to our sets of irrational numbers without them. All this tells us is that it is possible that the set natural numbers is bigger than that of the set of real numbers in (0,1) or that they are equal in size.
Now that we have cast doubt on Cantors diagonal proof, we can show how it is possible to find a function that does pair up the natural numbers to the irrational numbers. (the rational numbers can be paired, so we will ignore them for now)
We will generate a set of random irrational numbers between 0 and 1, our list might look like this.
0.5123453…
0.5125674…
0.5124195…
0.5127676…
0.5124295…
0.05127295…
…
Then we can imagine a deck of cards where the first card has the number 1 on it, the second card has a 2, third a 3 and so on, a stack with all the natural numbers in order.
The first number on our random list is 0.5123453… so we can pair it with the fifth card in our deck.
The second number on our list is 0.5125674… since we have used the 5 card, we must now use card 15. (the first two digits in reverse)
The third number on our list is 0.5124195… since we have used the 5 and 15 card, we must now use card 215. (the first three digits in reverse)
The fourth number on our list is 0.5127676… since we have used the 5, 15 and 215 card, we must now use card 7215. (the first four digits in reverse)
The fifth number on our list is 0.5124295… since we have used the 5, 15 and 215 card, we must now use card 4215. (the first four digits in reverse)
And so on.
So we have our pairing.
5 → 0.5123453…
15 → 0.5125674…
215 → 0.5124195…
7215 → 0.5127676…
4215 → 0.5124295…
50 → 0.05127295…
…
If you ponder on it for a while, you will see that there will never be an irrational number that we can’t find a natural number to pair with it.
We could sort the natural numbers and keep the irrational that it is paired with, together with it, and then look at the nth digit of the nth real number that it is associated with. For example,
1 → 0.14159265359798…
2 → 0.23606797749979…
3 → 0.3166247903554…
4 → 0.414213562373095…
5 → 0.5123453…
…
Diagonal = 0.13624…
then change the diagonal digits to something else and we would get an irrational number that is not in our list of irrational numbers.
Anti-Diagonal = 0.34333…
Since the number that we create is different from every number in our list of irrational numbers, it must be different at some finite point. (it is impossible for two irrational numbers to have the first infinite number of digits in common, then to be followed by some digits that are different.)
Since this diagonal defines a finite string of digits, that has not been seen on the right side of our set, then we know that same finite string does not appear on the left side either, since the left side is simply a reflection of the digits that have appeared on the right side. So we know that there exists a natural number equivalent to that string (or a substring of that string) that will be available to pair with the irrational diagonal number that introduced that finite string. Therefore, the cardinality of the two sets is the same.
And the continuum hypothesis is true, there is no infinity between the naturals and the reals.
You are mistaken in a few places here. First, you say that, in your process, "it will become harder to find a real number." It won't, really. You're making it artificially harder to find a new real number by generating a large pile of lists, but one cool thing about a pile of lists is that you can make them into one list. And, in fact, by pairing up reals with naturals, you've given us a really convenient method of doing this. Just order by the sizes of the naturals. Now we have one big list, and we can easily find a new real that is on none of the subset lists. No matter how many lists you add, you will never make it any harder to find a new real. It is thus never going to be possible to get all the reals onto your list/lists.
Now, let's look at the actual pairing you've generated. To break this pairing, let's stop looking at random irrationals, and start, y'know, picking them. Your pairing method should, after all, work for any order of irrationals, not just for weird orderings. The first irrational will be .1314159... This, obviously, pairs with 1 in your system. The next is .2314... And, in general, the nth element of this irrational ordering is a decimal point followed by the digits of n, followed by the digits of pi.
Here's the kicker. Using only this really specific kind of irrational number, I have eaten every single natural number. You say I will never find an irrational number that I cannot pair to a natural number? I pick .114142..., or one followed by the square root of 2. This number cannot be paired with 1, for 1 was paired with .1314..., it can't be paired with 11, because 11 was paired with .11314..., it can't be paired with 114, because 114 was paired with .114314..., and so on. This irrational will never find a partner before infinite digits, and natural numbers do not have infinite digits. Your pairing method does not work. No pairing method will.
Edgar Nackenson said “ It is thus never going to be possible to get all the reals onto your list/lists.”
True enough, but that is not the problem. The problem is that we have paired all the reals on infinite lists where each list has an infinite number of reals, with natural numbers and we still have lots of natural numbers left over. We have only used the natural numbers that have a 4 in the units place.
As for your second argument, let me use your own words to show you why it fails. We will consider the rational numbers instead of the irrational numbers.
Now, let's look at the actual pairing you've generated. To break this pairing, let's stop looking at random rationals, and start, y'know, picking them. Your pairing method should, after all, work for any order of rationals, not just for weird orderings. The first rational will be 1/1 This, obviously, pairs with 1 in your system. The next is 1/2 And, in general, the nth element of this rational ordering is a ratio of 1 divided by n.
Here's the kicker. Using only this really specific kind of rational number, I have eaten every single natural number. You say I will never find a rational number that I cannot pair to a natural number? I pick 2/3. This number cannot be paired with 1, for 1 was paired with 1/1, it can't be paired with 2, because 2 was paired with 1/2, it can't be paired with 3, because 3 was paired with 1/3, and so on. This rational will never find a partner before infinite digits, and natural numbers do not have infinite digits. Your pairing method does not work. No pairing method will.
Can you see how your argument makes the rational numbers uncountable?
Here's the kicker. It does not matter how many strategies do not work, we are only looking for one strategy that does work, and if we find one, then the cardinalities of the two sets are equivalent.
To the first point, it is incredibly irrelevant that you're leaving a pile of natural numbers in your back pocket. You can add as many other natural numbers as you want and you'll still fail to account for all the reals. We know this for a fact, because you already essentially used all the natural numbers. After all, the natural numbers with a 4 in the units place share a cardinality with all natural numbers.
To the second point, what you've said here does not function as a counterargument. The reason for this, straightforwardly, is that I haven't told you how the rationals are being paired with the natural numbers. Your mapping is fundamentally dependent on the order I present you with real numbers, which is why giving your mapping a specifically ordered pile of real numbers is a valid argument.
The pairing between naturals and rationals does not function in this way. Each rational is statically paired with a specific rational in a predetermined method, such that, if you name a random rational number, the system does not default to picking the first natural number, or even the first natural number of a given form, but rather a specific natural number determined by the system. It's a bit easier to see this with a more straightforward pairing, like odds to evens. The mapping here could easily be n -> n+1. So, if you name even numbers at random, say 8,614, I don't have to blindly grab at the first odd with a few properties. I can specifically tell you that this number pairs up with 8,613.
You are correct that finding individual strategies that fail to work is not sufficient, and that providing one counterexample would be more than sufficient. Proofs by counterexample are a beautiful thing, generating math truth easily and clearly. However, first, you have not provided a proof by counterexample. Your mapping doesn't work. This in itself doesn't prove that no mapping will work, but, second, something that does prove that no mapping will work is Cantor's diagonal argument. Simply take your mapping, reorder the natural numbers (preserving the mapping), and apply Cantor's diagonal argument, and you will have, again, missed 100% of the real numbers.
olavisjo Just checking, was there supposed to be a response here? Cause I got notified regarding one, and I dunno if it was deleted or is just weirdly invisible.
I can see it when I am logged into my account, but when I log out it is not there, so it would be "just weirdly invisible". I will post it again.
Can we segregate infinities of the "same" size by how fast the elements in the sets approach infinity? Meaning use "size" and "rate" to define different infinities?
No that concept doesn't apply to arbitrary sets. Not every set can be measured so the concept of "rate" doesn't apply. Some sets can't even be ordered so you can't even put them in a sequence that makes sense.
Excellent! Explicit coverage of bijection (correspondence) is often missed in other discussions of the hierarchy of infinities, though it is always implicit.
Vocabulary Wish-list: 1) Use "Aleph notation" when introducing the Tower of Infinities. 2) Bijection is "pairing", but is an instance of the more general topic of "mapping". 3) Integer/Whole numbers are "countable" (in principle) where Reals are not.
In general, deep understanding of a concept is less important that knowing the name of the concept (and its context), simply because we can Google for a name, but would have a tough time searching for a concept without knowing the related names and vocabulary.
In other words, a concept without metadata (name and context) is isolated from associated concepts. Math is already extremely abstract: Unlike, say, physics, where a specific physical phenomenon or famous experiment can be used as a touchstone.
Please use each video as an opportunity to not only examine a specific concept, but also to introduce all the relevant terminology that would make Google more useful to viewers desiring to learn more.
Hmmm. There may be a future Infinite Series video here: How to use Google to learn more about a math concept.
TBH, I'm glad she avoided some of the lingo around sets, I think it can be a bit distracting when approaching this topic for the first time. This stuff can be pretty mindblowing to the uninitiated, and I think it's prudent to break their brains slowly. :)
I definitely agree with #2 though, it would be nice to have had a more thorough definition of 'bijection' here, emphasizing it's not just being able to match one way.
1-1/(x+1) is also a bijection of the interval from 0 to 1 and 0 to infinity.
I've a question. we know that the distance between two physical points on earth namely x0 and x1 is real infinite. And we also know that the difference between two point of time namely t0 and t1 is real infinite too. And we know that we can travel from x0 to x1 in (t1-t0) time with a constant speed. Is that means division of two real infinites may be a natural number or we are not actually moving and living in inception world (!)?
Would the numbers between 0 & 1 have the same cardinality as the Natural Numbers?
What numbers between 0 & 1?
Depends, between 0 and 1 are:
Countably infinite rationals
Uncountably infinite reals
Unsetly infinite surreals
What if you define the size of a set by the sum of all the elements? What do the infinities look like?
Thanks gurl. I learned a lot of math (8 semesters just from calculus) and never could grasp the continuum hypothesis. But now I think I do.
My only question is, is there an alef omega infinity? An infinity which as infinitely many infinities smaller than it?
Great video; I hope you shed some light on the axiom of choice, which is the main reason why Continuum H. is unprovable
AC is not the reason. It will still be unprovable without AC.
Is the set of complex numbers bigger than the reals? Since they form a plane, i'd have to use a hemisphere to do the trick at 4:33.
They are the same size. These "tricks" get much trickier and technical when it comes to more interesting bijections. Simply put, it is possible to fill a space of any dimension with a single curve (line which can turn) which is naturally one-dimensional.
Space-filling curves are overkill. For every real number you can take just the even numbered digits to make a real part and the odd numbered digits to make an imaginary part.
Martin Epstein That doesn't work though. There is this catch that real numbers are notexactly isomorptic to infinite decimal expressions.
Take following real numbers and look where your function maps them into complex plane:
0.09090909... -> 0 + 0.9999i
0.10000000... -> 0+1i
While the real numbers on the left are distinct, the complex ones on the right are identical, so it is not a bijection.
Jakub Pekárek Sure, but the same is true of space-filling curves.
source: math.stackexchange.com/questions/43096/is-it-true-that-a-space-filling-curve-cannot-be-injective-everywhere
I was really just going for a surjective map. By the Schroder-Bernstein theorem, if you can get a surjective map and injective map separately then you've proven the existence of a bijective one (I only learned that just now).
Source: en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem
Martin Epstein What you give here talks about continuous mappings, which is different object alltogether, we are considering set to set (no structure) map.
The Cantor-Bernstein theorem would work for a different setting. One way is simple as R is contained in C. The other way could be made through inverse of what you described, but it's trickier. You'd basically need to establish that the inverse construction creates distinct infinite decimal expression. Then you would need to prove that distinct obtained expressions correspond to distinct real numbers, which would be a real pain as it is not true universally. And then you could invoke the theorem.
However, if you want to invoke theorems, I'd suggest saying that C = RxR while for infinities X and Y it holds |X|*|Y| = max{X,Y} (in this case even withou AC). Thus simply |C| = max{|R|} = |R|.
Also, proving |X|*|Y| = max{X,Y} is simple, certainly simpler then doing stuff described above.
Nevertheless the original question was geometrically oriented, thus I used curve.
You definitely explained it better than others. Still just a weird concept to me. What are the applications?
Math tends not to be concerned so much with application as opposed to just finding stuff out. Oftentimes results like this are fairly 'useless' in the most obvious sense, their value is not in the result, but in the things used to prove that result. She didn't go over the proof directly in the video, but part of the proof that the Reals are larger than the Naturals is an *incredibly* useful tool for proving things called "Cantor's Diagonal Argument." This argument is a clever way of constructing new items in a set from old, such that you can show no possible bijection can exist. This is normally a pretty hard thing to do. This technique of 'diagonalization' shows up in a lot of places from Analysis to Topology and elsewhere.
In addition, the concept of 'pairing as counting' (bijection) is a useful tool for another important approach to mathematics, reduction. Bijection as a tool can help us establish which sets are related, and translate problems from one set to another by means of that bijection. Often, it's valuable to do that because what might be difficult to solve in on domain (i.e., in one Set), is very easy to solve in another. Even if it's not easy enough in that first set, we can perhaps more easily solve it in another set we know bijects on to it, and so on. It lets us take a path through various mathematical environments to eventually arrive at something that makes our problem soluble.
Hope that helps.
jfredett It definitely does and I appreciate the in-depth response. Hopefully I'll learn more about some of that in the following years so that the concepts as a whole make more sense. It seems like something I could really get in to. Thanks again.
+Majin Krocket I'm glad it helped. Math is really a very different sort of field, it's equal parts abstract nonsense and concrete, useful tools for understanding the world and how it works. If you're self-studying (heck, even if you're not) there's a lot of good material around the internet for learning about the various fields in detail, +3blue1brown is a *phenomenal* channel for survey level stuff about math, highly recommend checking him out.
jfredett I just recently found 3blue1brown about a month ago but have watched channels like numberphile, vsauce, vihart, etc. for a long time. I mentioned learning it over the next few years because I'm going to do a math minor for my CPE degree. I'm really excited for some of the more in-depth classes.
Like already said, bijections are incredibly important everywhere in maths. In maths you are always interested wether you can see two objects you defined as the "same". You do that by using bijections which have special properties depending on the objects you are looking at. So bijections are in a way the weakest type of "equality" of objects, the equality of sets concerning how many elements there are.
The most usefull application of knowing which sets have the same size is probably saying that these to objects can't have a bijection of any type between them, so especially the ones you are looking for.
Is the infinity between 0 and 0.1 the same as the infinity between 0 and infinity?
Yes, any bounded interval of reals can be mapped one-to-one with the set of all positive real numbers, or even with the set of all real numbers. But you may have been asking a different question. Can an interval of reals be mapped one-to-one with natural numbers? And to that the answer is no; given any function from natural numbers to real numbers (or to an interval of reals), there is some element of the latter set which the function doesn't cover (and, by extension of the same proof, the set of uncovered numbers can be mapped one-to-one with real numbers).
I want a full version of the intro music. That beat is fire.