A Hierarchy of Infinities | Infinite Series | PBS Digital Studios

[@pbsinfiniteseries]

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.

[@Cashman9111]

logicians count 0 as natural number and analysts don't, it makes sense if you think about it

[@amdenis]

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.

[@akhil3279]

hi just whated to ask you something what if the so called rule of bijection dont apply to infinite sets..

[@Nixitur]

+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".

[@jamez6398]

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...

[@BaryLevi]

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.

[@r75shell]

There are vast of topics. I guess she could talk whole time only about infinity :D.

[@AxiomTutor]

She says there's a link in the description to a rigorous proof, presumably that's Cantor's.

[@pbsinfiniteseries]

Totally! The whole series could be about infinity! Y'all have great suggestions for thing you'd like to see in follow-up videos

[@alexandrugheorghe5610]

+ imaginary and complex planes?

[@jpphoton]

He said next week. :)

[@Ermude10]

"In english, this means that..." Haha, I've used this phrase too when explaining concepts after a formal definition!

[@fishtrick1_]

lol

[@benjaminpedersen9548]

I just realized that the definition of cardinality actually relates really well to the Pigeonhole Principle.

[@BeCurieUs]

Man I hope you go into the incompleteness theorem more!

[@DekuStickGamer]

^ This. I'd love to see a Infinite Series video on that, with cool animation and shit.

[@BeCurieUs]

DekuStickGamer Ya, it is one of those areas I can't hear enough about because it is just so....odd.

[@David_Last_Name]

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.

[@enlightedjedi]

I'd like it, too!

[@BeCurieUs]

***** Gödel escher and bach would be a good place to start.

[@pomtubes1205]

Avg. length of each infinite series vid is about 8 min. 8 is like infinity standing up.

[@adityakhanna113]

Standupmaths?

[@SimonClarkstone]

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.

[@BrknSmmtr]

Lemuel Ogabang This is so nerdy and great that I have to reply, but I don't know what to reply with except this.

[@naimulhaq9626]

What about 24.

[@apple54345]

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

[@DavidIngerman]

"There're infinitely many sizes of infinities." Are there countably or uncountably many different infinities?

[@NutziHD]

Uncountably many

[@prateekgargx]

Equivalent to continuum hypothesis tho

[@supriyoset3289]

If you got enough time, then yes, countable. 😅

[@thomasbernhardqed]

fun fact: both

[@nzqarc]

Much much more than either

[@DontMockMySmock]

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?

[@mscottveach]

+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.

[@KaiKunstmann]

Cantor's diagonalization proof only proves that the infinite list of real numbers is not finitely complete, which is kind of pointless.

[@mscottveach]

+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?

[@MuffinsAPlenty]

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.

[@mscottveach]

+Muffins aha.. in other words, i can safely ignore.

[@trampleguy]

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

[@AliJardz]

I didn't understand this when vsauce or vihart explained it, but I get it now! Thanks!

[@jancoker2699]

V’s always make things confusing

[@romanski5811]

+Jan Coker Yeah, he should just stop doing what he does. He shouldn't confuse people.

[@jasonmelvin9102]

@@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

[@zubmit700]

Love this channel! Keep up the good work!

[@792p]

I had to watch the video many times over to truly appreciate the writing. Awesome stuff !

[@sasikumarannandakumar6148]

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.

[@brucea9871]

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.

[@guitarheroprince123]

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.

[@flymypg]

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.

[@adriel3000]

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.

[@groinBlaster31]

Ever since finding this concept in like, first year math, I found it fascinating!!

[@while_coyote]

Are the complex numbers larger than the real number or the same size?

[@zairaner1489]

They are the same size

[@while_coyote]

Are there larger infinities?

[@DaviddeKloet]

The class of all sizes of infinities is so big, it doesn't even fit in a set!

[@while_coyote]

What's another example other than the two given in this show?

[@pbsinfiniteseries]

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.

[@YYYValentine]

Vsauese's video "count past infinity" is an other awesome video of the subject.

[@ekadria-bo4962]

Bálint Áts but he actually to much talking rather give a argument.

[@hemalatasamavedula7345]

Bálint Áts the vsauce video is better than this one.

[@cauchyh3879]

Vsauce videos are non educated and just for 'show' like common drama.

[@Zyxxi]

No mention of aleph?

[@zuccx99]

Sad.

[@pspicer777]

@@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.

[@johnrflinn]

No, it too Borges.

[@johnrflinn]

ooops (sic) it's

[@christianrayfield4365]

Just finished by linear algebra final, now I finally get to relax while learning math : ))))

[@MrHowbout]

How did it go? I loved linear algebra!

[@christianrayfield4365]

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. : )

[@AxiomTutor]

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.

[@christianrayfield4365]

That makes a lot of sense, haha.

[@MelindaGreen]

I thought there was a natural ordering of the infinities which is that each higher infinity is the power set of the previous one.

[@sligocki]

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.

[@MelindaGreen]

Shawn Ligocki What would such a fractional infinity even look like?

[@sligocki]

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.

[@MelindaGreen]

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.

[@truebaran]

No, it does not follow even if you assume continuum hypothesis. Your statement is the so called generalized continuum hypothesis.

[@giorgioyoung102]

What a fantastic series! Thanks for helping me explain to my family what I love and want to spend my life doing!

[@Kram1032]

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.

[@mza3764]

not really Many important mathematical results are related to the specific ZFC model

[@thetruecookie8115]

The explanations are really clear. I actually understand the Video's context!

[@abhdya]

Is there any proof that the natural number infinity is the smallest? Or is that just assumed?

[@jaredgraham4022]

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).

[@ninadmunshi2879]

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.

[@pbsinfiniteseries]

Great suggestions! Thanks

[@alexalden5249]

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?

[@b43xoit]

It was described as an open interval. So the endpoints are excluded.

[@therealoldnosey8689]

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?

[@Tumbolisu]

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.

[@iankrasnow5383]

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.

[@therealoldnosey8689]

Ian Krasnow What sets definitely end?

[@iankrasnow5383]

Finite sets

[@therealoldnosey8689]

Ian Krasnow some examples?

[@xeuszzz]

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!

[@luisvasconcellos4549]

What about complex numbers?

[@pbsinfiniteseries]

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?

[@luisvasconcellos4549]

I'm gonna think about it a little bit. By the way, the videos are awesome and the content is extremely interesting, congratulations.

[@guest_informant]

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?

[@112BALAGE112]

Luis Vasconcellos The Hilbert curve does exactly that. A bijection betwen 1D and 2D space.

[@martinepstein9826]

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.

[@ZanderzMcCluer]

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.

[@r75shell]

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.

[@cauchyh3879]

Without evil, you can't even calculate the area of a circle. we need the evilness

[@hosonlam1452]

Yes. We don’t need this concept in mathematics with computer today. All are finite and human being is keep advancing ahead!

[@hosonlam1452]

Cauchy Riemann In fact, the area we got was always not right but just approximation only.

[@cauchyh3879]

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

[@hosonlam1452]

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.

[@kichigan1]

Thank you Kelsey. Amazing show.

[@kj01a]

She didn't even get into surreal numbers, because she didn't want to get brain matter all over your keyboard.

[@benjaminprzybocki7391]

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.

[@larrytroxler7017]

Glad she covered the continuum hupothesis! it's fascinating how it could true or false, but it doesn't matter!

[@jayskies8]

that gap tho :o

[@ryanmike9833]

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?

[@sebij6811]

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).

[@wwguee3338]

This video is infinite confusion

[@jonatanguitar]

0:57 It took literally less than 1 minute to blow my mind this time!

[@zeewirszyla]

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.

[@christophersewell6611]

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.

[@chrissidiras]

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?

[@sciencehour9476]

Where do you get that type of background music?

[@MuhsinFatih]

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

[@EneriGiilaan]

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'.

[@MuhsinFatih]

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

[@EneriGiilaan]

Agreed that this is a quite unintuitive - but still 1st year university math. For a relatively decent explanation search for KZbin video: Infinity is bigger than you think - Numberphile.

[@MuhsinFatih]

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 ^^)

[@Altoclarinets]

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.

[@pbsinfiniteseries]

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.

[@soranuareane]

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.

[@JubilantJerry]

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

[@soranuareane]

Clever. Thank you.

[@bessermt]

"All Brontosaures are thin at one end, much much thicker in the middle, and then thin again at the far end." - Anne Elk

[@RSLT]

I LOVE THIS CHANNEL ❤❤❤❤❤❤❤

[@filipve73]

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)

[@vedantbhutra1118]

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)

[@pbsinfiniteseries]

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!

[@vedantbhutra1118]

Oh yeah, that makes sense. Thanks a lot!

[@mina86]

Check out Hilbert’s hotel. Or if you want something more hard-core, Vsauce’s video on the Banach-Tarski paradox.

[@vedantbhutra1118]

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?

[@mina86]

Yep.

[@1JDRM]

This is an excellent explanation of cardinality. Amazing job!

[@edwardsizemore8340]

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

[@Tazirai]

Mind Blown. Totally loved this.

[@gJonii]

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

[@philipoakley5498]

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]

Well we know that 1.0 is EXACTLY equal to 0.99999..., they are the same number

[@philipoakley5498]

@@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.

[@farkler4785]

@@philipoakley5498 I’m not following how any of this relates to cantors diagonalization theory. And the “we” is mathematicians as a whole

[@saikat93ify]

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.

[@-_Nuke_-]

Just subbed due to this amazing video and the clarity in which it was presented! Great work

[@mistafabro]

The intro music to this channel is kind of badass

[@sebastianstuwe3523]

hi ive got a question r the real numbers the last step? i mean r imaginary numbers as infenite as real numbers or natural numbers? i ask cause my math teacher told me that numbers like i dont belong on the numbers line they r more likly 2 be somewhere above or under the line so they its like they would be another infinety like the space between 1 and infinety or the infinety between 1 and 2 so the first infinety is defined by 1 infinety the second grade of infinety is defined by 2 infineties and finaly my question is if the imagenary numbers form a third grade of infinety wich includes 3 infineties as explained inb4

[@b43xoit]

Those are (at least) two different questions. The reals are not the last step, because another step is the set of all the subsets of the reals. But as for the imaginaries, there is clearly a bijection between them and the reals. As for the set of complex numbers, other commenters (above) point to a proof that the size of that set is the same as that of the reals.

[@briand8090]

What could you subtract from infinity to result in a natural number once you have reached infinity? Is it possible to add the inverse of any real or imaginary number to result in a natural number? Would infinity plus negative infinity still be infinite in set size?

[@musicalBurr]

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.

[@onionpsi264]

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.

[@jamiebarnes4820]

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.

[@frustumator]

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.

[@onionpsi264]

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

[@jamiebarnes4820]

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.

[@jamiebarnes4820]

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.

[@janpahl6015]

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)

[@janpahl6015]

thanks a lot

[@sergiogarza2519]

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!

[@thomaskn1012]

If there are infinite sizes of infinities, what are examples of others besides the two groupings demonstrated in this video?

[@yashcherivirala]

Hey just saw this video today. Not sure if you are still answering questions on this. But what about the complex number set which has all real numbers and numbers like square root of -1 and so. Can we identify a bijection between Complex numbers and Real numbers ? Or is the complex number set a bigger infinity than the real number set ?

[@MuffinsAPlenty]

The complex number set is indeed in bijection with the real number set. The easiest way to do this is to prove that the complex number set is in bijection with ordered pairs of real numbers (where a+bi is paired with (a,b)), and that the set of pairs of real numbers are in bijection with the set of real numbers. There are many different, but tricky, ways to prove that R is in bijection with R^2. I don't know of any "direct" proofs of this fact (in that they don't ever seem to explicitly establish an actual bijection, but rather use set theory facts to show that a bijection must exist). For example, using cardinal number arithmetic, you can prove that the cardinality of R and R^2 are the same: |R^2| = |R|^2 = (2^(ℵ0))^2 = 2^(ℵ0*2) = 2^(ℵ0) = |R| Or some people use the existence of space filling curves to show that there exists a surjection from R to R^2. And we also know there exists an injection from R to R^2. A set theory fact is that if there exists an injection and a surjection from one set to another, then there exists a bijection between them as well. And there are other proofs too.

[@MagicGonads]

What if you define the size of a set by the sum of all the elements? What do the infinities look like?

[@tabamal]

Did the introduction of so many infinities into mathematics give rise to any useful conclusions?

[@GargaGaming]

Asking the real questions ^

[@Ennar]

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).

[@guest_informant]

3:44 *What about the Continuum Hypothesis?* Doesn't the CH _basically_ say: It can neither be proved or disproved that there is not an infinity in between the natural numbers and the real numbers? Edit Googling, Wolfram says: _The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small" infinite set of integers aleph_0 and the "large" infinite set of real numbers c (the "continuum"). Symbolically, the continuum hypothesis is that aleph_1=c. Problem 1a of Hilbert's problems asks if the continuum hypothesis is true._ _Gödel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory. However, using a technique called forcing, Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory. Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the axiom of choice)._ _Conway and Guy (1996, p. 282) recount a generalized version of the continuum hypothesis originally due to Hausdorff in 1908 which is also undecidable: is 2^(aleph_alpha)=aleph_ _(alpha+1) for every alpha? The continuum hypothesis follows from generalized continuum hypothesis, so ZF+GCH|-CH._ _Woodin (2001ab, 2002) formulated a new plausible "axiom" whose adoption (in addition to the Zermelo-Fraenkel axioms and axiom of choice) would imply that the continuum hypothesis is false. Since set theoreticians have felt for some time that the Continuum Hypothesis should be false, if Woodin's axiom proves to be particularly elegant, useful, or intuitive, it may catch on. It is interesting to compare this to a situation with Euclid's parallel postulate more than 300 years ago, when Wallis proposed an additional axiom that would imply the parallel postulate (Greenberg 1994, pp. 152-153)._

[@AniconicArden]

would a prime number infinity be larger or smaller than real or natural numbers?

[@nzqarc]

The infinity of prime numbers is the same as natural numbers

[@EBukhari]

Can you please speak more to the other models of infinity towers and what infinities they place in-between Uncountable and Countable infinities?

[@MikeRosoftJH]

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.

[@mydroid2791]

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?

[@prototypeinheritance515]

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.

[@JuusoAlasuutari]

Is the cardinality of the set of primes equal to the cardinality of the natural numbers? It would make sense because the primes can be paired with their indices (which are natural numbers).

[@HeavyMetalMouse]

What sort of axiom(s) would need to be added to ZFC, such that they don't create inevitable contractions with known theorems, but would allow Cantor's conjecture to be proven or disproven?

[@ossiebird0]

So are there more curves then there are points on a straight line?

[@Lexivor]

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.

[@ossiebird0]

Thanks, I'm really going to have to think about your answer.

[@insidetrip101]

Your proof for the intervals being of the same size as the real numbers seems a bit ambiguous. When you drew the semi circle, the end points of the semi circle would have a slope that is paralell to the real numberline and therefore the rays from the center of the semi circle would never intersect with those two points (the numbers at the end points). So my question is whether or not a closed interval is of an equal infinity of an open interval. I'd really appreciate a response!

[@robtminn1]

Are there infinite dimensions of infinity? For example, (2inches)^2 = 4 inches ^2, so it has 2 dimensions, x and y. So would infinity inches ^ infinity have an infinite number of dimensions?

[@taschke1221]

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.

[@TGC40401]

Would the numbers between 0 & 1 have the same cardinality as the Natural Numbers?

[@b43xoit]

What numbers between 0 & 1?

[@nzqarc]

Depends, between 0 and 1 are: Countably infinite rationals Uncountably infinite reals Unsetly infinite surreals

[@bens4446]

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.

[@SalameeQueijos]

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"

[@cyberneticqualanaut7207]

What are some great textbooks on this subject?

[@Rik079]

Is it possible to state that if the cardinality of infinite(natural) can be described as x, the cardinality of infinite(real) is x^2? (As any interval of natural numbers contains infinite real numbers, for example the interval [2, 5] wich has 3 * infinite real numbers in it, so the interval [-infinite, infinite] has infinite * infinite (infinite^2) real numbers in it)

[@nzqarc]

Actually it isn't x^2, it's 2^x

[@vpr1422]

The background music is beautiful.

[@RayHorn5128088056]

Does The quantification of the infinite render them finite?

[@kenwalter3892]

Is the infinity between 0 and 0.1 the same as the infinity between 0 and infinity?

[@MikeRosoftJH]

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).

[@jaredhoeft2832]

Is there a class of infinity larger than the one that the natural numbers/etc fall into?

[@zairaner1489]

Yes the ral numbers for example

[@megauberduber]

So what happens if you subtract the sum of all natural numbers from the sum of all real numbers?

[@PapaJefeYT]

I want a full version of the intro music. That beat is fire.

[@Doping1234]

The continuum hypothesis confused me a bit. Could it be false and disproven by an example of an infinity between N and R resp. true and a case of Gödels incompletenes theorem OR does the concept of an infinity between N and R not make sense within the ZFC theory?

[@markzambelli]

I was about 15 when I encountered Aleph Null, Beth...etc... Mind blowing stuff (in a book in Leeds University library)

[@candanbolukbas]

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 (!)?

[@mancinellismathlab7451]

I'm surprised there was no mention of the rational and irrational numbers? Rational numbers as a dense subset of the reals is quite amazing

[@cr-gn6rr]

10 seconds in and my brain already exploded

[@darthsavage4025]

Is the size of the set of imaginary numbers the same as the real numbers?

[@ciCCapROSTi]

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?

[@bjorng9867]

awesome, great channel with a lot of potential in sense of added value (looking at the CC, PBS, CGP grey landscape).

[@iankrasnow5383]

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".

[@amdenis]

I have a question. You define the Natural numbers as 0, 1, 2... and you say that evens and natural numbers (counting numbers) are the same size, adding zero to the set that includes 2, 4, 6.... However, I had learned that the Natural numbers are actually 1, 2, 3.... (no zero, as that was what I understood as "whole numbers"), meaning that the infinite sum in a physical construct as defined via the Zeta Riemann function series sums to -1/12, with a cardinality of aleph null. Am I missing something, as I have little doubt you have a broader and deeper math background than I do?

[@pseudorandomly]

As the pinned comment at the top explains, set theory (the realm of this video) counts 0 as a natural number, where other areas of mathematics may not.

[@Corvaire]

Ah, no mention of Omega Omega.. Corvaire is disappointed. :O(-

[@RovingPunster]

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).

[@b43xoit]

In the '70s they used "bijection".