Become a Creator today!Start creating today - Share your story with the world!
Start for free
00:00:00
00:00:01
56: More Sheep than You Can Count (Transfinite Cardinal Numbers)
433 Plays4 years ago
Look at all you phonies out there.
You poseurs.
All of you sheep. Counting 'til infinity. Counting sheep.
*pff*
What if I told you there were more there? Like, ... more than you can count?
But what would a sheeple like you know about more than infinity that you can count?
heh. *pff*
So, like, what does it mean to count til infinity? What does it mean to count more? And, like, where do dimensions fall in all of this?
Ways to support the show:
Patreon-Become a monthly supporter at patreon.com/breakingmath
(Correction: at 12:00, the paradox is actually due to Galileo Galilei)
Distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For more information, visit CreativeCommons.org
Music used in the The Great Courses ad was Portal by Evan Shaeffer
[Featuring: Sofía Baca, Gabriel Hesch]
Recommended
Transcript
Introduction & Episode Overview
00:00:00
Speaker
Look at all you phonies out there. You posers. All of you sheep counting. Counting till infinity. Counting sheep. Pfft. What if I told you there was more there? Like more than you can count? But what would a sheep like you know about more than infinity than you could count? Pfft. You probably don't even know about infinity. So like, what does it mean to count until infinity? What does it mean to count more? And like, where did dimensions fall in all of this?
00:00:27
Speaker
All this and more on this episode of Breaking Math. Episode 56, more sheep than you can count.
Breaking Math Poster & Purchase Info
00:00:38
Speaker
I'm Sophia. And I'm Gabriel. And this is Breaking Math. Let's start off with some plugs. Yeah, the plug that you often hear very frequently, because it's so worth it, is our Breaking Math poster, which is essentially a 30,000 foot view of the entire field of tensor calculus.
00:00:55
Speaker
which is used a lot in non-Euclidean mathematics, especially in Einstein's relativity, in general relativity. In fact, there's even a section on the poster that has that, so it's an awesome gift. What excites me about this poster, and I think I've said this before, is I used to go to the bookstore of my Alma
00:01:15
Speaker
modern and they had all of these tutorial little card foldouts that essentially had all of the major concepts and everything from trigonometry to calculus one to to everything even finance yet they only had those for the introductory and undergraduate courses this is kind of like that but for a graduate level physics and math course at least that's how I like to think of it
00:01:35
Speaker
Yeah, it's pretty cool. If you like Einstein, if you like vectors at all, if you like cool illustrations that look kind of sci-fi, you might want to spend $15.15 plus $4.50 shipping and handling in the United States for a tensor poster. And you could get that on our Facebook poster store, which is at facebook.com slash Breaking Math podcast and you just click on shop.
00:01:54
Speaker
Let's say you want to support us monthly. If you support us for $1 or more a month, that comes out to, what, like $0.03 a day. You could have free episodes. I mean, ad-free episodes. You could have the outlines that we use for the episodes, which have the stuff organized pretty well. If you support us with more than pi to the E dollars, which is $22.46,
00:02:14
Speaker
We'll get you a poster too. And now we're allowing people from overseas to print the poster themselves, just so we have a look, because we want to share the goodness with everyone. Yeah, and actually that's, I actually really like that. I mean, you know, that's gosh, I know that some people may advise against doing that, but you know, so now if you live overseas, you also have access to our poster, just send us an email and it will work out those arrangements for how exactly
Engaging with Breaking Math
00:02:40
Speaker
we do it.
00:02:41
Speaker
The message just on the patreon or email us and what's our email again? It is breaking math podcast at gmail.com Yep Twitter handle at breaking math pod and contact us on both of those if you have ideas for episodes questions about episodes We're pretty we're on there pretty frequently trying to build more of a relationship with you the audience Yes, we also have a website that needs dire updating but has some pretty cool apps on it called breaking math podcast dot app and
00:03:07
Speaker
Yes, that a P P. So just go to HTTP colon slash slash breaking math podcast dot a P P. And there is a 3d interactive hypercube that you can modify and add more or fewer dimensions. I think that's the coolest thing in the world. I just think that you need a song to go with it. Like maybe in the gada de vida or something, you know, we could have the hypercube song, not just a cube.
00:03:32
Speaker
Yeah, I don't know it's like it it's all it's mesmerizing in the same sense that like a lava lamp is you know, but oh, yeah No, I have I have embarrassingly stared at my own code for like like 20 minutes one time because I was like undulator So you can literally go there right now and with your controls on a desktop. Unfortunately, it doesn't work with a phone yet but we're getting all everything worked out with our domain and
00:03:55
Speaker
Yes. All the things with your keys, you can add dimensions. You can have a five dimension or six dimension hypercube and change the colors too. It's really cool. So check it out. Check it out. And I know that this is a, I don't mean to take the plug section too long, but if you have a moment, we'd really appreciate you seeing our sister show on YouTube.
00:04:14
Speaker
We have a guest who has made an appearance on a few episodes named Dr. Alex Alaniz and he has a physics and math YouTube show that I'm also a part of called the touring rabbit holes podcast named after Alan touring and we currently have I think between four and five hundred subscribers So it's kind of a bit of a small channel, but we are putting our whole heart into it So if you could check that out give us a like and subscribe. We would love that too. Thank you very much
Cardinal Infinity & Uncountable Sets
00:04:37
Speaker
So what is what what is what are we talking about today? We're talking about Cardinal infinity is obviously right. Yes. Yes Now you you said it infinities the I love the title if I may real quick The title of the show is more sheep than you can count. Obviously, that's an allusion to infinity. I love it. It's a fantastic
00:04:55
Speaker
And the reason why we chose that is because there are sets of things, collections of things that cannot be counted. And, um, and I mean that in very real sense, like not even all the real numbers, right? Like w not real numbers, all the natural numbers, right? One, two, three, four, five, et cetera. You can count all of them. Theoretically they are called countable because you can put them all in a list that has, um, the first item at the first place, the second item at the second place on and on and cover all of them.
00:05:25
Speaker
There are some sets you cannot do that with, and one of those sets is the set of real numbers. And we'll show why on here, but basically, you cannot list all the real numbers in any way, shape, or form. You will always leave things out if you have any list, no matter if it's infinitely long.
00:05:41
Speaker
There's some really nuanced facts about infinity and that we've realized in the field of mathematics in the last 200 years. One thing I love about this episode though is I got especially interested in this topic when I taught sixth grade math and what's cool is we're going to get into this in a little while but we had to do these Socratic seminars at the school I taught at. It was a charter school.
00:06:05
Speaker
And the kids started exploring infinity completely on their own. And they had these open group discussions that got into this. And what I found wonderful is there is absolute overlap between some of their discussions. I think one of the topics that was brought up was, does a circle have the same amount of points as a sphere?
00:06:26
Speaker
And, you know, yes, I kid you not that we had sixth graders talking about this in a Socratic discussion. Not everything was maybe, you know, correct in that discussion. But a lot of most of it touches on the kind of cardinal infinities that we're going to talk about today. For example, the answer to the question of does a circle have the same amount of point as a sphere? Surprisingly is yes. And we will find out why. Yeah, yeah, absolutely.
00:06:52
Speaker
So Gabriel, tell us a little bit more about the Socratic discussions and we'll go along and kind of talk about how each concept maybe relates to Cardinal Infinity. Oh, I'd love to. Okay, yeah, again, this part excites me because I've never ever done something quite like this in a school.
00:07:07
Speaker
Uh, the school I taught at, um, you guys can Google search it. It actually has won a lot of awards. The school is called Cottonwood classical preparatory school. A little shout out to my old school, uh, where I thought I taught out for a while and it is a charter school with, with some, and as a charter school, they do things a little bit differently.
00:07:24
Speaker
Um, you know whether it's curriculum or or even the you know, the time of day that they start with well with this charter school One of the things they wanted to include was uh, Socratic dialogue so written into the charter every single class whether it's history or humanities or an art class or even even Mathematics, uh, there was a time I think we tried for every two weeks roughly didn't always happen, but that was the original
00:07:47
Speaker
every two weeks on a Friday in our math class we would move all the desks out of the way and move all the chairs into sort of an oval so to speak and there would be a subject that was either introduced by a student or it could even be a quote or perhaps something you know if you're just getting started off the teacher might but really it's quite student-led.
00:08:08
Speaker
There would be a quote on the board and the students just had to discuss it kind of in a popcorn style and it's open-ended. So in a mathematics course, there was a lot of philosophical quotes that we found on numbers. I think we found some quotes from Frederick Nietzsche on numbers. It would always be like a quote from Nietzsche or other philosophers that had something to do with mathematics and the students would explore it.
00:08:29
Speaker
During one particular session, a question came up and I don't remember who it was from, but the question arose. Does a circle have zero angles or infinity angles? What a fabulous question. Initially, I think when I was younger, I would definitely say zero angles.
00:08:46
Speaker
But as the discussion went, you know, I mean, can you see how somebody might say infinity? Oh, yeah. I mean, I thought about this a lot when I was a kid, too, because I mean, you look at the progression from triangle to square and you notice it looks more circle like and the angles get shallower and shallower. And so this question of how many angles does a circle have?
00:09:06
Speaker
You could almost say infinity. I mean, yeah. And the question though is like, which type of infinity too? Cause we're going to talk about, we're going to talk about countable and countless infinities. And it turns out that a circle has an uncountable infinity of different points. And if each point has an angle that can be associated with it, then you can definitely say that a circle has infinite angles. Yeah.
00:09:26
Speaker
Now, there's more relevant things that came about from this rich, rich conversation on this day. And again, as I said earlier, it was a dialogue with sixth graders. So not everything was rigorously defined, but that misses the beauty of it, I think.
00:09:42
Speaker
I mean I'm just touching on the fact that these questions do have a Like real ways of looking at them and because they are such common questions. I mean these sixth graders had them I had them at an early age. Yeah, I think it's good to go into Their answers. Yeah in real quick
00:10:00
Speaker
So in this episode, we're going to discuss the history of countable and uncountable and introduce some terminologies. But before we do that, I just want to say a few more things that came up during that conversation. Not all of them, but a few more things.
00:10:15
Speaker
It was amazing. I had kids come to me the week after and say, Mr. Hesh, I was thinking about last week's conversation over the weekend and I think that in some ways zero and infinity are kind of the same thing. And I was just like, wow,
Functions & Infinity Examples
00:10:28
Speaker
this is crazy. Which is a very common mathematical intuition as well in a cool way. Like mathematicians, you look at something, a pattern that occurs with zero and one, a good place to look next is infinity.
00:10:39
Speaker
Yeah, and just all these like fascinating things show the human intuition and the human strive for Infinity and I think that what it comes from is I mean if I eat breakfast every day I can imagine very easily eating breakfast forever There's I mean like the word drip and dribble are related in old English in this similar way to drip it means to drip once to dribble means continuously drip and
00:11:05
Speaker
So the idea of continuity, I mean, I don't think it's any coincidence that differential and integral calculus would have to do quite a bit with the concept of this kind of continuity also deal with infinities.
00:11:20
Speaker
Alrighty, now before we get into a few deeper concepts, we will review ordinal infinities versus cardinal. So ordinal numbers are for assigning unique numbers to collections of sets and things like that. Yeah, so if I have two copies of all numbers next to each other and I want to give them a unique address, I have to use ordinal infinities.
00:11:39
Speaker
However, we know that they have the same amount of, the same amount of items, and this amount that we're talking about is the cardinality. So the cardinality is like, how big is a set? And so here's an interesting thing, an interesting paradox, and I think it's due to Gauss, I believe. Is Gauss or Hilbert? I can't quite remember. I'll put that in the show notes. But the number of integers is the same as the number of squares.
00:12:07
Speaker
That is wild. Oh, so that like grates on one's intuition, at least I think it was. Yeah, right, because you seem like you're skipping things. It's like, how can they have the same number? And the reason why is because every number, one squared is one, right? Yes. Two squared is four. Yep. Three squared is nine. Every number has a square and every square is different. Correct. Okay. Oh my goodness. That's still wild because one can also say truthfully that in any finite set,
00:12:35
Speaker
Oh, no, that's not true. I was going to say, you can erase that too. In any finite set, there's fewer squares than there are, but that's... Oh, yeah. No, yeah. It only works on infinite sets. And that's where we get like this weird, like these weird paradoxes. So we need to define things in a way that is a little bit more concrete. And so that's the way we say that two sets have the same cardinality. If there is a function that is both injective and surjective, then maps one to the other. So let's talk about what a function is really quick, right?
00:13:04
Speaker
And we're also going to talk about later what exactly injective and surjective mean, because those are very important terms here. Yeah, so let's say that the function that we're dealing with is somebody's social security number. It's a number in the United States that's used to uniquely identify people.
00:13:21
Speaker
So, the domain of the function that maps a person to their social security number will be the domain of all people, right, in the United States. The co-domain is the set of all social security numbers. So, what is injectivity? Injectivity means that it maps different elements from its domain to different elements of its co-domain. So, let's take this function, right, this social security function. If two people are different, will they have different social security numbers?
00:13:46
Speaker
That means that the set is injective. And the same thing happens with the squares, right? If two numbers are different, then their squares are different. Now let's talk about surjectivity. It's kind of the opposite of injectivity. So a set is surjective if every element of its codomain, so what's being mapped to, has some element that maps to it from the domain. So let's look at our social security function. If we have a social security number, is there a person to go along with that number?
00:14:12
Speaker
Yes, there is. Meaning that the that the function is surjective. So if a function is both surjective and injective, it's kind of a one to one and onto thing, right? Yes. Meaning that they have the same number of sets. And that works for the square function, because the square function, if you map every number 123 to squares, you're not going to skip a square, right? Correct. So that is how you tell if two numbers have the same cardinality, you create a function that maps between them.
00:14:38
Speaker
Okay. Yeah. Very good. And again, so you say injective, meaning that each person will map to one unique social security. Yeah. Like if two people are different, then their social security numbers will be different. Okay. And that means injective. And surjective means that for every social security number, there's at least one person to go along with that. Okay. And there's one unique as well, right? No, surjective doesn't have to be unique, but injective plus surjective means it's unique. Oh, got it. Okay. Very good. That's why both conditions have to exist. Okay.
00:15:06
Speaker
All right, now we'll go into a little more depth into cardinal in affinities specifically.
00:15:11
Speaker
So let's say we want to have a symbol, right, for the cardinality of the set of all natural numbers. That's a cardinality of 1, 2, 3 through infinity, or like 1, 2, 4, I mean 1, 4, 9, 16, you know, all the squares. Anything countable all has the same infinity. Anything countably infinite is what it's called. And it has, and the term for this is aleph not, which is a little aleph symbol, which is a Hebrew character with a little zero written next to it. And that is a cardinality of the set of all natural numbers.
00:15:40
Speaker
So how can we show or not show that the cardinality of natural numbers is equal to or not equal to the set of all real numbers? I'd say an inductive proof where you... I don't know. Well, there is an inductive proof for it, but I'm getting at the idea that we would have to show that a function exists that is both injective and surjective to show that they have the same cardinality or show that no such function exists to show that they have different cardinalities.
00:16:08
Speaker
Very good. So we're going to have to prove a couple of lemmas. The space from zero to one, right? I claim in the real numbers has the same cardinality as a space from negative infinity to positive infinity. And that's your lemma. I love that word lemma. That's like also a claim, so to speak. Yeah, it's like a mini proof.
00:16:25
Speaker
Cool. And then here we have the proof to your lemma that the space from zero to one has the same cardinality as a space from negative infinity to positive infinity. And we start by identifying a function, right? Yep. And the, the function that we, uh, that, uh, that it's honestly arbitrary, but one function that works is LN of X over one minus X quantity.
00:16:44
Speaker
which is negative infinity at zero, zero at one-half, and positive infinity at one. Let me say that one more time. One of the functions is the natural log of the quantity, it's a fraction, x over one minus x. Okay, I'm following. Yeah, and this function happens to be strictly increasing.
00:17:05
Speaker
No, we know that if it's strictly increasing, we know it must be injective, right? Because that means that every number will have a different number associated with it. And we also know that it has every number from negative infinity to positive infinity because it's a continuous function, and we see that it approaches negative infinity at zero, zero, one-half, and positive infinity to one, so it sweeps all the way through all the natural numbers, all the real numbers.
00:17:29
Speaker
And so since this function exists, we know that those two have the same cardinality. So now any claims that we make about the cardinality of the space between zero and one can be made about the cardinality of all real numbers, which actually does us a big favor.
00:17:42
Speaker
And, um, so now we're going to do a second lemma. Let's say I have a set of real numbers between zero and one. So they all start with zero dot, right? Zero point. Sure. And they have an infinite number of digits after them. They can, a lot of them can be zeros, but there's an infinite number cause they're real numbers, right? Okay. Sure. Like PI or whatever. Yeah. Maybe one's well, yeah. Like PI minus PI minus three, you know, like they're all numbers between zero and one. Sure.
00:18:05
Speaker
So I claim, and this is a second lemma, that if we have a set of these, right, no matter how big the set is, we could always find a number that is not in that set. Okay. And all we have to do is change one digit at a time. Okay. So what we do is we create a number that changes the first digit of the first number to something else, because then it won't be the same as that number, right? Correct. Just like add 0.1 or add any arbitrary change to it.
00:18:35
Speaker
Yeah, like change 0.13592 to 0.23592. And so the second number might match this new number that we're creating. However, we're not going to let it because we're going to take its second digit. So let's say it's 0.123. We're going to take that 2. We're going to change it to something. It doesn't matter what we change it to. We can change it to 3. But all we have to do is we have to change it to something that won't match the second number. OK.
00:18:58
Speaker
And we could do that to every number right in the set. And this new number that we come up with will not be any of those natural numbers, any of those real numbers. So that means that any set that we have, we could choose a real number that isn't in that set. Okay. Wow. And therefore that has some significance then for real numbers.
00:19:16
Speaker
Yeah, because what it means is that the set of all real numbers is uncountably infinite, because you can make this thing arbitrarily large. So that means that any countable set of these things, which is like what it means to be able to go through them one at a time, you'll always be able to generate a number that is not within the set, no matter how many numbers you have.
00:19:33
Speaker
Yeah, yeah, and that is the proof that for that first of all the number of real numbers is uncountable Mm-hmm and second of all that the infinity is infinitely greater than Lf naught Okay, we call this cardinality Lf 1 Wow, so let me let me read these three bullet points again if I may yeah So as you said earlier just to sum this up the set of all real numbers is uncountably infinite and this proof here is
00:19:57
Speaker
It says, since you can always make a new number from any set of numbers, that means that any countable or countably infinite set of real numbers cannot contain every number. That's crazy. Even though it's infinite. Even though it's infinite. Wow. Yeah. Which means that we cannot put the real numbers in any order that will fill up. We will always have gaps.
00:20:17
Speaker
Yeah, because if that were the case, then a new number could not be made from them, which is a contradiction. And since the space from zero to one has the same cardinality as the real numbers, that means that the real numbers are uncountably infinite. As you said earlier, that's because we have an injective and surjective map between them, right? Yes. Yes. Oh, wow. Goodness. Mind blower. I know. It's fun, though, right? Yeah. Yes, fun. So fun. Yes.
00:20:41
Speaker
But yeah, that's the proof and it's called a Cantor's diagonalization argument and it was created in like I think the 1870s Actually, it was 1891
00:20:53
Speaker
Okay, very good. And we have another one called space-filling curves. So here's the thing. We have Lf naught, which is the cardinality of the natural numbers or the fractions, right? Yes. Any one of those, because you could count all the fractions as we showed in the last episode. And then Lf1 is the cardinality of the real numbers, right? Yes. So you might think that there's some Lf2 that's the cardinality of the plane, Lf3 for 3D space, and so on.
Dimensions & Cardinality
00:21:24
Speaker
Well, Cantor, our friend Cantor, in 1879, proved actually that the plane, the 3D space, 4D space, any dimension of space has the same cardinality as the real numbers. Wow. And the way that he did it, he did it in 1879 with the infinity hotel argument, which we talked about in the last episode for a different reason. But in 1890, he created a discovery, which is used today in like some research actually called space filling curves.
00:21:54
Speaker
to answer this question. And what a space-filling curve is, is imagine you scribble on a piece of paper, right? And let's say that we, and let's say the scribble cannot touch itself, right? And let's say we can, let's say actually instead of a scribble, it's like yarn that we're pushing onto a piece of paper. And theoretically, we could actually have the yarn be so twisty that it actually goes through every point. Now, since it goes through every point,
00:22:21
Speaker
It means that it is a surjective function, right? And since every point has a unique point associated with it, it is injective. Meaning that there's an injective and surjective map between the unit line and the space-filling curve.
00:22:37
Speaker
Meaning that the plane and the line and the cube all have the same cardinality, which to me is pretty mind blowing. Wow. Oh my goodness. Okay. Say that one more time. Hold on a second. So when you say cardinality, the points that are contained.
00:22:54
Speaker
Yeah, it was basically what I'm saying is you can squiggle through every there's a way that has been defined to squiggle through every number in the space where X and Y are between zero and one and so basically since you can squiggle through all that with a space filling curve you can
00:23:12
Speaker
It has the same cardinality as the real numbers, because there is a one-to-one and onto function, a surjective and injective function. Wow, that's pretty mind-blowing. Goodness, that's using some formal terminology, but that's still pretty wild.
00:23:28
Speaker
Now, I'm glad that we prefaced this next section on the discussion on space-filling curves, because the conversation that transpired with my sixth graders during that Socratic dialogue on circles and infinity touched on a lot of these things. And I obviously I don't think at the time anyone realized it. This section we call sphere weirdness. On our outline. Yes. And we'll discuss a few other things here. Let me see here.
00:23:56
Speaker
So during the conversation on circles and infinity, we started off by talking about whether a circle has a zero or an infinite amount of angles and we entertained the idea that it has an infinite amount of angles. There were other discussions as well that happened both during that Socratic seminar and the weeks following with students who discussed with me during my office hours that I'm now aware of. So one of the students had said, imagine if you have two circles and these circles are empty circles.
00:24:26
Speaker
Yeah circles on a piece of paper right or some kind of 2d plane, right? Correct And you know, so yeah, so we say that they're hollow I suppose two-dimensional circles, but they're hollow two-dimensional Circles and those are also known as one spheres because their surface is one-dimensional and their surface is all we're caring about right now Yeah, yeah and the discussion with something like this imagine two of these circles and you're holding them in your hand and they're flat when they're not touching this is
00:24:53
Speaker
You know obvious, you know, they they touch at no points, but when they um touch they touch at one point their tangent at one point Yeah, when yeah when they when they touch when they barely first start touching. Yes. Yeah, like they they they kiss if you were to kiss The circles it's at one point now if you were to overlap the circles partially not all the way but partially how many points do two circles partially overlap at two
00:25:19
Speaker
Right? Yeah, exactly. Yeah, like you've got your, you now have two points, you know, you can even draw straight lines through them, you don't have to, but you have two points. Now, if the circles are the exact same size, if you keep moving the circles until they perfectly overlap, one circle perfectly overlaps on the other, it looks like you only have one circle, how many points do the circles overlap at?
00:25:41
Speaker
And we actually have an answer with this now. Yes. And here's the thing, because the points that they overlap at, if you take a circle, right, and you peel it, unpeel it, it's like a straight line, right? Correct. Meaning that it has the same cardinality as a straight line. Correct. Because any homeomorphism like that will imply usually a one-to-one and onto function. Yep. And so you could say that there are aleph one points that they overlap at. The cardinality of their overlap is aleph one. Correct. So it goes from zero to one to two to aleph one. Yes.
00:26:10
Speaker
Yeah. Yeah. Now, of course we didn't use the term aliph one in that dialogue. We just talked about, you know, some students said, Oh, is it an infinite point? Or how about just we say all points. And during the conversation, we agreed that all points was the preferred term as it's not scary. Like infinity is scary. Yeah.
00:26:25
Speaker
And here's that thing too, though, is that like, once we have these concrete ways of dealing with infinities, like Lf9 and Lf1, instead of just like infinity being like this nebulous faraway thing, we get more comfortable with talking about the infinite. And I think that's a cool thing about math is being able to parcel these incredible concepts into like tiny symbols. Yeah. And it's also me, you know, applauding this approach of a Socratic discussion, you know, where it's not, it's graded on participation points, you know, but we didn't grade it on like correctness. So I think it's a really cool thing.
00:26:54
Speaker
Now, the conversation gets weirder. Just hang tight, because it gets much, much weirder. To review to where we are now, we start with two circles, and you know, they're empty. They're not solid circles, they're just flat circles. Apart, they touch at no points. Touching, one point. Overlapping, two points. Completely overlapping, LF1, infinity points, or all points. Now, let's try this again, but with this time, with spheres. If you have two spheres that are empty, and what are those called?
00:27:23
Speaker
Two spheres because their surface is two-dimensional and we only care about the surface. Yeah, so it's not filled like a bowling like a billiard ball It's empty like a it's like a shell so you have two spheres and they don't touch at no points at all they touch at one point like if they kiss then if they partially overlap ah Here's the question if you have two spheres that partially overlap how many points do they partially overlap at?
00:27:50
Speaker
And you'll notice about this is that those two circles will overlap at a ring that is where they touch, right? Yes. And we've discussed earlier, a ring has Lf1 points, correct? Yep, that's correct. Now, and I know where you're going with this, the two spheres are now going to completely overlap, correct? Yes, two spheres that are the same size completely overlap at how many points?
00:28:11
Speaker
Yeah, and here's the thing, because we can unpeel a sphere, stretch it out so that it's on the 1 by 1 square, we know that that has the same number of points as LF1, right? Correct. So the sequence here goes that the spheres touch at no points, then they touch at one point, then they touch LF1 points, and then they continue touching at LF1 points. Even though there's an infinite number of these rings that they now touch at, it is still the same cardinality.
00:28:36
Speaker
That blows my mind. That absolutely blows my mind. It's just insane. So, you know, and it's interesting because they're even different dimensions, you know, when the two spheres partially overlap and they form that ring or that that 2D flat circle, if you will, that's that's an infinity. But then when they completely overlap, you know, your intuition strongly says that's, you know, more of them is now overlapping, you know.
00:28:57
Speaker
And I think what's also interesting, too, is that if we get three spheres together, right? If you two overlap slightly, then you have a ring. And then you have a third two sphere intersecting with that. Now you have two points that they intersect at. And if you add a fourth sphere, there's one point that they intersect at. And that, of course, is how GPS works. The three spheres are how long it takes for a light to reach you from three to many satellites. And the fourth sphere is the Earth.
00:29:24
Speaker
Yeah. Wow. Fascinating talk. Okay. So yeah, so that got pretty intense pretty quickly. And I don't think the conversation went much further than that, that discussion. Although there were questions in general, things like, you know, we, we, we talked about it earlier. Um, if, if we're talking about points and lines and things like that, and we talked about dimensionality, one of the questions was, um, if, if, if a line can be one dimensional is a dot zero dimensional or is nothing zero dimensional, does
Zero-Dimensional Space & Points
00:29:51
Speaker
that make sense?
00:29:51
Speaker
Yeah, in this case, I'd say that dimensionality has to be attributed to something that exists. However, something can exist with no length or breadth, and that's called point, you know? Yes, yes. There's an entire field of mathematics that we looked at for this episode called, what was it? A nil-dimensional space. A nil-dimensional space, and that's where some of these terms come from. And what does it say about like a single point? That's a zero-dimensional circle?
00:30:17
Speaker
Yeah, the point can be visualized as a zero dimensional, zero dimensional hyperball. And remember, a ball is like a filled in sphere. And you know, it's interesting, though, about the difference between a zero ball and a one and a negative one sphere is that a zero ball is technically the inside of a negative one sphere, or like, or you could say that a negative one sphere is the outside of a zero ball.
00:30:38
Speaker
However, the measure of a zero ball is one, because there's one point, so we just say one. Yes. However, the measure of a negative one sphere is zero. And I think the reason why is because there's nothing outside of a zero ball in zero-dimensional space. There's room for exactly one thing. Yes. But the formulas just bear this out. And I just bring this up because it does speak to your point earlier that you'd had with your sixth graders on how zero and infinity are bedfellows.
00:31:04
Speaker
Yes, yes, absolutely now that that's touched on a lot in the book called zero the biography of a dangerous idea by charles sheath That is on the favorite book list of many many many of my math professors in college again That's zero the biography of a dangerous idea by charles sheath But also I think a lot of the the topics we touched on are involved in the book flat land Which is the short story from when?
00:31:27
Speaker
Like late 1800s Victorian era. Yeah, yeah, and they also talk about a single point where you have one one occupant is that single point that is both the God and the only occupant at all and it's only aware of himself.
00:31:39
Speaker
Yeah, and it is really fascinating, all this stuff. There's so many things that could be done on infinities. This is not the end of our discussions on infinities, the end of these last two episodes, but just the fact that you have these ways of talking about infinite numbers, and also such that ordinal and cardinal versions are different, I think is just mind-blowing.
00:32:03
Speaker
Yeah, and also, you know, oh gosh, and this also gets into trying to conceptualize things like derivatives and other aspects of math. Great conversation. So I guess if we were to conclude this episode, I recognize that it's very difficult to, you know, finalize these conversations, but I very much promote the Socratic Seminar method in mathematics classes, science classes, and other classes as well.
00:32:28
Speaker
And I do find it interesting your thing of one of the points that I saw that you made and is that that with your students is that if you have two spheres of different sizes like one is a little sphere like a tennis ball and the other one is a really big sphere like one of those exercise balls. Yeah.
00:32:44
Speaker
Type size do they have the same number of points or different number of points? And the reason why they have the same number of points is because you can you could have a latitude and longitude on each one, right? Yes, and each latitude and longitude can map to unique latitude and longitude on the other one, correct? So that function the latitude and longitude function that transfers between the two is both one-to-one and onto it's both injective and surjective. Yes, and yeah, injective just means one-to-one and surjective just means onto and
00:33:11
Speaker
Yeah, but um, yeah, I just think I I think it's fascinating these infinities that happen outside of lists these infinities that happen in the fibers between things and Yeah, and I believe there are actually greater infinities than this one But that's gonna have to take have its own episode if we talk about it
00:33:31
Speaker
He dealt with some of the weirdness of these concepts as well as the math that goes into proving them. But one thing we hope you got out of our discussions of infinity is this. The term infinite is like the term etc. You have to know what you're talking about to be talking about anything at all. I'm Sophia. And I'm Gabriel. And this has been Breaking Math. Email us at breakingmathpodcast at gmail.com with any comments or questions we love hearing from you. Poster facebook.com slash breakingmathpodcast. Click on store if you want it, it's cool.
00:33:59
Speaker
Yeah, and we would love you know, if you would love to support us another way we said earlier is to go to the great courses plus comm slash breaking math They're one of our sponsors and there are a group that we're very proud to be teamed with because they're very educational as well I'm very happy that breaking math has been able to Extend these kind of offers and I mean that sincerely Till next time