WEBVTT - The Infinity Hotel 0:00:03.120 --> 0:00:06.000 Welcome to Stuff to Blow Your Mind from how Stuff 0:00:06.000 --> 0:00:13.119 Works dot com. Hey, you're welcome to Stuff to Blow 0:00:13.160 --> 0:00:15.800 your Mind. My name is Robert Lamb and um Julie Douglas, 0:00:15.840 --> 0:00:18.479 and this is our second episode on infinity. If you 0:00:18.520 --> 0:00:20.560 missed the first one, if you recommend you go back 0:00:20.600 --> 0:00:22.160 and listen to that one. But we are going to 0:00:22.239 --> 0:00:24.119 do a little recap here in case any of the 0:00:24.160 --> 0:00:28.000 information has fallen out of your head in the time being. Yeah, 0:00:28.040 --> 0:00:31.080 so let's talk about what infinity is and what it 0:00:31.160 --> 0:00:34.440 is not, because that's our nice little baseline there. It 0:00:34.600 --> 0:00:37.680 is not a real number, because we discussed before. You 0:00:37.720 --> 0:00:40.599 can't say that infinity is X number. You can't write 0:00:40.640 --> 0:00:42.640 it on a check. You can use a symbol, I guess, 0:00:42.640 --> 0:00:46.240 but good luck cashing it. Uh, why have I never 0:00:46.320 --> 0:00:50.600 done that? Uh? It cannot be measured. Of course, that 0:00:50.840 --> 0:00:52.880 is tied back to it not being a real number. 0:00:53.360 --> 0:00:56.960 It's not something that's growing, it's not doing anything. It's 0:00:57.000 --> 0:01:02.000 the ultimate Zen concept. It just is yes. And of 0:01:02.000 --> 0:01:06.479 course this has been an important topic to philosophers, uh, theologians, 0:01:07.000 --> 0:01:09.840 great thinkers throughout time, and that's one of the primary 0:01:09.840 --> 0:01:13.080 things we discussed in our last episode because Ultimately you 0:01:13.120 --> 0:01:17.119 get into discussions about the boundless, about the limits of things. 0:01:17.200 --> 0:01:21.000 I mean, you're dealing with everything from the basic human experience. 0:01:21.720 --> 0:01:23.480 You know, how do I deal with the fact that 0:01:23.680 --> 0:01:26.120 my life only stretches on until so so long? How 0:01:26.160 --> 0:01:27.480 do I deal with the fact that I can only 0:01:27.480 --> 0:01:31.120 perceive the universe with a certain bubble of perception. And 0:01:31.120 --> 0:01:34.039 then you start imagining a god and you're trying to 0:01:34.160 --> 0:01:37.280 figure out how that works. Is God is God? Infinite? 0:01:37.400 --> 0:01:41.160 Is God physically infinite? Is God infinite in quality as 0:01:41.160 --> 0:01:44.640 opposed to quantity. It's to say the biggest idea is 0:01:44.680 --> 0:01:47.440 it is the the the infinite idea that the humanity 0:01:47.520 --> 0:01:51.000 has tackled with over the ages. Yeah, and we talked 0:01:51.040 --> 0:01:54.440 about George Tvorski writing for Ion nine, and he used 0:01:54.520 --> 0:01:58.320 the chess board to illustrate this idea, saying that all 0:01:58.360 --> 0:02:01.800 the other pieces having number assigned to it except for 0:02:01.840 --> 0:02:07.840 the king, whose number value is infinite. Yeah, that chess 0:02:07.920 --> 0:02:10.760 is It is interesting because we mentioned before, you have 0:02:10.840 --> 0:02:14.400 a finite number of playable games for chess, and then 0:02:14.480 --> 0:02:17.760 you then you have this king that has an infinite 0:02:18.120 --> 0:02:21.119 importance within the game, So you have two different types 0:02:21.160 --> 0:02:23.480 of infinity going on there. Yeah, you lose, you lose 0:02:23.520 --> 0:02:26.200 the king, You lose the game because zer or some 0:02:26.240 --> 0:02:27.960 game if you get rid of the king, and so 0:02:28.000 --> 0:02:30.639 again it ties back to this whole idea about eternity 0:02:30.680 --> 0:02:35.320 even consciousness uh embedded in there with infinity. So of 0:02:35.360 --> 0:02:40.079 course it becomes with this great uh media area for 0:02:40.240 --> 0:02:44.280 philosophers to really tussle with, because it's got the afterlife there, 0:02:44.280 --> 0:02:47.720 it's got the idea and the question of God, and 0:02:47.840 --> 0:02:51.040 by its very nature, it's an idea that we cannot 0:02:51.080 --> 0:02:55.160 fully contain and we cannot really completely grasp it, and 0:02:55.240 --> 0:02:57.880 yet we try to and so did philosophers as we 0:02:57.919 --> 0:03:00.760 discussed the last episode, and they used to try to 0:03:00.880 --> 0:03:03.440 prove this out or try to make their arguments even 0:03:03.480 --> 0:03:06.320 more persuasive. So today we're gonna look at some examples 0:03:06.440 --> 0:03:10.079 in math and physics to see how well we can 0:03:10.320 --> 0:03:14.959 contain the idea of infinity and reach some sort of meaning, 0:03:15.440 --> 0:03:17.800 which we already talked about as sort of impossible, like 0:03:18.000 --> 0:03:22.720 meaning is infinite in itself for various reasons. To just 0:03:22.760 --> 0:03:27.480 to recap mathematics and infinity, you voluencountered numbers like twelve 0:03:27.560 --> 0:03:32.520 point one to nine, eight, seven, six, five, three, two, one, zero, one, 0:03:32.680 --> 0:03:34.960 just on on and on forever. You just have to 0:03:35.040 --> 0:03:37.920 stop counting those numbers. After a while, you have to 0:03:37.920 --> 0:03:41.680 stop recording them because it becomes a pointless exercise. You 0:03:41.800 --> 0:03:44.880 just round up, right. Is that because humans are finite 0:03:45.000 --> 0:03:49.680 and we kind of continue to count infinitely. Yeah, that's 0:03:49.720 --> 0:03:53.560 that's Uh, that's kind of the big topic of discussion here. 0:03:53.760 --> 0:03:57.160 And so you kind of have two schools of thought 0:03:57.200 --> 0:04:00.960 in mathematics, right when when it comes to to struggling 0:04:01.040 --> 0:04:02.880 with the infinite, and it kind of comes down to 0:04:02.920 --> 0:04:06.760 some of the core arguments with mathematics itself. So is 0:04:06.760 --> 0:04:11.360 is mathematical infinity? Is this a mere human creation, an 0:04:11.360 --> 0:04:14.200 extrapolation into the world where there's no truth? Or does 0:04:14.280 --> 0:04:18.200 mathematical infinity actually exist? Well, this comes down to the 0:04:18.279 --> 0:04:21.839 question of God to rite. There are some observable things 0:04:21.920 --> 0:04:26.280 of how infinity works in mathematics, um, and I say observable, 0:04:26.279 --> 0:04:30.039 but you can it's just as we discussed in your example. 0:04:30.120 --> 0:04:32.919 There numbers can go on and on forever, and we 0:04:32.960 --> 0:04:37.840 have finite lives. Uh, we can't see God in Therefore, 0:04:38.000 --> 0:04:40.960 some people would say he or she or it does 0:04:41.000 --> 0:04:44.080 not exist. So you're grappling with some of the same 0:04:44.120 --> 0:04:47.920 concepts in math and in infinity. So if you were 0:04:47.960 --> 0:04:52.600 to ask the constructivists or the institutionists, they would say 0:04:52.640 --> 0:04:55.560 that classical mathematics, well, they deal with the sort of 0:04:55.560 --> 0:04:58.640 math that God would do. An infinite God he can 0:04:58.680 --> 0:05:02.599 tackle with infinite numbers. Uh, and and and the idea 0:05:02.640 --> 0:05:05.840 of a mathematical infinity. But we are humans, we are finite, 0:05:05.880 --> 0:05:09.280 and we would rather focus with human mathematics with finite 0:05:09.440 --> 0:05:12.400 numbers because that ultimately is what relates to what we do. 0:05:12.440 --> 0:05:15.000 You know what I hear with that, my brain hurts. 0:05:15.080 --> 0:05:18.479 I'll just I'll just leave it to God. He or 0:05:18.520 --> 0:05:20.560 she or it can deal with it. Yeah, I mean, 0:05:20.560 --> 0:05:24.719 it's kind of the idea that infinity is an interesting 0:05:24.720 --> 0:05:27.720 topic to tackle with as an abstraction, but does it 0:05:27.839 --> 0:05:30.680 really relate to the work that I'm doing here, the 0:05:30.720 --> 0:05:33.880 work that mathematics needs to do in the world. And 0:05:33.880 --> 0:05:36.800 then they're the formalist Now. Formalism is a theory that 0:05:36.839 --> 0:05:39.600 holds that statements of mathematics and logic can be thought 0:05:39.600 --> 0:05:42.479 of as statements about the consequences of certain certain string 0:05:42.520 --> 0:05:46.839 manipulation rules. Um systematic formulation of the concept of mathematical 0:05:46.880 --> 0:05:50.400 formulism arose directly as a reaction to the paradox discovered 0:05:50.440 --> 0:05:54.120 within set theory, which studies the concept of infinity. According 0:05:54.120 --> 0:05:59.200 to twenty century mathematician Abraham Robinson, infinite totalities do not exist, 0:05:59.320 --> 0:06:03.559 either really or ideally. Any mentioned of infinite totalities is meaningless. Still, 0:06:03.600 --> 0:06:06.960 the business of mathematics must continue as if infinite totalities 0:06:07.000 --> 0:06:10.240 actually do exist. Right, we said, there are a couple 0:06:10.240 --> 0:06:12.480 of things that we're talking about. A system here in 0:06:12.480 --> 0:06:16.200 this idea that you can again capture infinity. You can 0:06:16.320 --> 0:06:21.760 define it um and then set theory, which is something 0:06:21.920 --> 0:06:25.440 that a man named Georg Cantor came up with. And 0:06:25.480 --> 0:06:29.120 he's sort of the rabble rouser here. According to Natalie 0:06:29.200 --> 0:06:32.760 while cover writing for Quantum magazine, infinity was boxed and 0:06:32.839 --> 0:06:35.960 sold to the mathematical community in the late night concentury 0:06:36.000 --> 0:06:40.360 by the German mathematician Georg Cantor, who invented a branch 0:06:40.440 --> 0:06:45.039 of mathematics dealing with sets collections of elements that ranged 0:06:45.160 --> 0:06:48.560 from empty the equivalent of the number zero to infinite. 0:06:49.000 --> 0:06:52.200 So Cantra stepped in there really and began to define 0:06:52.320 --> 0:06:57.520 infinity in a much more specific way. Yeah, and uh 0:06:57.560 --> 0:06:59.840 he was he was a one of these individuals who 0:06:59.880 --> 0:07:03.120 was hated as well as love, depending on who you're asking, 0:07:03.200 --> 0:07:06.520 because he was bringing up some really mind blowing, disturbing 0:07:06.760 --> 0:07:13.320 takes on infinity. Yeah. Natalie Walquiver says that other mathematicians 0:07:13.360 --> 0:07:18.680 initially despise what they called this mess of infinities. And 0:07:18.680 --> 0:07:21.200 we'll talk more about this because it's largely due to 0:07:21.280 --> 0:07:24.760 his ideas of set theory, and one of his colleagues 0:07:24.880 --> 0:07:28.600 called them a grave disease. Another called him a corrupter 0:07:28.960 --> 0:07:33.120 of you. Uh. Sadly, he was so vilified that he 0:07:33.080 --> 0:07:37.600 actually um fill into a lifelong depression after that. Um. 0:07:37.640 --> 0:07:39.920 But I think it tells you how radical some of 0:07:39.960 --> 0:07:43.040 these ideas were at the time and still are really, 0:07:43.800 --> 0:07:46.680 because what it does is it just sets minds ablaze 0:07:47.200 --> 0:07:52.160 as to what what reality is and isn't. Essentially, Yeah, 0:07:52.160 --> 0:07:54.000 I mean, one of the core concepts to come out 0:07:54.000 --> 0:07:56.840 of it is that there are different kinds of infinity 0:07:57.000 --> 0:07:59.800 and some are bigger than others. Well, that that's thinking 0:07:59.840 --> 0:08:01.760 for a second. Yeah, let me let's get into the 0:08:01.760 --> 0:08:05.000 math sweats here. I'm having the math sweats right now. Uh. 0:08:05.080 --> 0:08:11.600 Set theory is essentially a useful language for describing mathematical objects. 0:08:11.720 --> 0:08:14.600 And what we're talking about is a nine item list 0:08:14.680 --> 0:08:18.600 of rules that Cantor came up with called Zermelo Frankel 0:08:18.720 --> 0:08:21.880 set theory with the axiom of choice or the z 0:08:22.200 --> 0:08:26.240 f C, established and widely adopted in math by the 0:08:26.320 --> 0:08:30.320 nineteen twenties. Okay Um. One of the axioms says that 0:08:30.400 --> 0:08:34.120 two sets are equal if they contain the same elements. 0:08:34.800 --> 0:08:38.520 Another simply asserts that infinite sets exist. So you can 0:08:38.559 --> 0:08:41.280 have these infinite sets that are one to one equal 0:08:41.280 --> 0:08:43.600 to each other, and you have them going on forever 0:08:43.640 --> 0:08:45.880 and ever and ever and ever. And Cantor showed that 0:08:45.920 --> 0:08:48.640 for any infinite set, forming a new set made of 0:08:48.679 --> 0:08:52.280 all the subsets of the original sets represents a bigger 0:08:52.440 --> 0:08:57.360 infinity than that original set. So once you have one infinity, 0:08:57.400 --> 0:08:59.480 you can always make a bigger one by creating a 0:08:59.520 --> 0:09:02.400 subset the original set, and then an even bigger one 0:09:02.440 --> 0:09:04.320 by making a set of all the subsets, and so 0:09:04.400 --> 0:09:07.000 on and so forth. And there are an infinite amount 0:09:07.040 --> 0:09:10.840 of infinities in different sizes. And he also proved doubt 0:09:11.040 --> 0:09:13.360 that an infinite set of even numbers like two, four, 0:09:13.400 --> 0:09:15.560 or six could be put in that one to one 0:09:15.559 --> 0:09:19.880 correspondence with all counting numbers like one to three. So 0:09:20.080 --> 0:09:22.760 that came to this idea that there are as many 0:09:22.840 --> 0:09:27.520 evens as there are odds and evens in an infinite set. 0:09:28.000 --> 0:09:32.760 There's one problem with this, though, real numbers, yes, like 0:09:32.840 --> 0:09:34.840 the one that you talked about earlier. You know that 0:09:34.920 --> 0:09:38.000 for example point zero zero zero one or pie right 0:09:38.040 --> 0:09:39.719 three point one four or so on and so forth, 0:09:39.760 --> 0:09:41.960 blah blah blah blah. They go on for an ever 0:09:42.160 --> 0:09:45.640 and they are uncountable, and they don't correspond in a 0:09:45.679 --> 0:09:48.040 one to one fashion with counting numbers. But hold on 0:09:48.080 --> 0:09:50.080 because we will get back to that. If that didn't 0:09:50.080 --> 0:09:54.400 thinking completely. Will return to the unccountability of real numbers 0:09:54.640 --> 0:09:57.520 in a minute, once we have a structure in which 0:09:57.559 --> 0:09:59.800 to house them. All right, I am going to dab 0:10:00.080 --> 0:10:03.920 at the infinite number of sweat molecules dripping off of 0:10:03.960 --> 0:10:06.520 me right now. Uh No, it's not too bad actually, 0:10:06.840 --> 0:10:09.040 Uh And let's take a break. When we get back, 0:10:09.559 --> 0:10:19.360 you'll talk more about the infinite hotel. If you're out 0:10:19.360 --> 0:10:21.280 there and you're thinking, I don't know what they're talking 0:10:21.320 --> 0:10:23.640 about there, they're losing me. I'm sinking beneath the waters 0:10:23.720 --> 0:10:26.120 here in the It is the in the deep end 0:10:26.280 --> 0:10:32.679 of the mathematical, philosophical, theological, physical pool. I need help, 0:10:32.720 --> 0:10:35.079 while we need help to for the love of infinity, 0:10:35.120 --> 0:10:37.920 give me an example. Yes, we do have, We do 0:10:37.960 --> 0:10:40.680 have an example. We have a flotation device. And it 0:10:40.760 --> 0:10:43.120 is a thought experiment, because the thought of experiments, as 0:10:43.160 --> 0:10:47.320 always they can take some very difficult to grasp ideas 0:10:47.360 --> 0:10:51.960 and concepts and put them into into a metaphorical form 0:10:52.000 --> 0:10:55.160 that we can latch onto and then and then actually 0:10:55.160 --> 0:10:59.319 explore the concept a little more easily and sometimes a 0:10:59.360 --> 0:11:02.520 little a little deeper. Right, And what could be more 0:11:03.320 --> 0:11:06.840 of a form inaccessible than a building, which is essentially 0:11:06.880 --> 0:11:10.200 what we're talking about. Um, this is an infinite building, 0:11:10.240 --> 0:11:13.760 an infinite hotel. And this was created by German mathematician 0:11:13.840 --> 0:11:17.800 David Hilbert who was obsessed with cantors work and and 0:11:17.920 --> 0:11:20.880 came up with us little ditty to try to explain 0:11:20.920 --> 0:11:23.920 a hotel with an infinite number of rooms. All right, 0:11:24.160 --> 0:11:26.520 it's so it's a hotel, infinite number of rooms? How 0:11:26.559 --> 0:11:28.440 many how many rooms? Does it have to like an 0:11:28.480 --> 0:11:33.560 infinite amount? Oh but accountable infinite amount, which is going 0:11:33.600 --> 0:11:35.520 to turn out to be important. All right, So let's 0:11:35.559 --> 0:11:38.440 say just to just to roll out the basic entry 0:11:38.520 --> 0:11:42.000 level portion of the thought experiment. I show up to 0:11:42.160 --> 0:11:45.640 the Infinity Hotel, all right, and and I need a 0:11:45.720 --> 0:11:48.439 room for the night. I know there's a convention in town. Uh, 0:11:48.559 --> 0:11:52.439 the local set theory convention maybe, and uh, and I 0:11:52.640 --> 0:11:54.120 I know that all the hotels are booked, but I 0:11:54.120 --> 0:11:56.800 really need a room. So the Infinity Hotel sounds like 0:11:56.800 --> 0:11:59.080 a good place to go, all right, So let me 0:11:59.559 --> 0:12:04.480 a my jaunty little manager's bellhop cap here. What I 0:12:04.520 --> 0:12:07.160 am going to do is, I'm going to ask the 0:12:07.240 --> 0:12:10.120 guests in room do you have vacancy? Always? We have 0:12:10.160 --> 0:12:12.920 an infinite amount of rooms. Okay, I see a lot 0:12:12.920 --> 0:12:15.839 of cars in the parking lot. Hold up, okay, just 0:12:16.840 --> 0:12:19.040 look at a magazine or something. We'll figure this out. 0:12:19.160 --> 0:12:21.280 I'm going to ask the guests in room one to 0:12:21.480 --> 0:12:24.480 move to room two, and then the guests in room 0:12:24.520 --> 0:12:27.440 two to move to room three, and so on and 0:12:27.480 --> 0:12:32.600 so forth. Every guest moves to room N plus one. 0:12:33.240 --> 0:12:36.080 And since there are an infinite amount of rooms here, 0:12:36.640 --> 0:12:39.280 then there's room for every guest to move into a 0:12:39.320 --> 0:12:41.920 different room. So you're saying that even though they're already 0:12:42.040 --> 0:12:46.600 infinite guests at the infinity hotel. Occupying the infinite rooms, 0:12:46.880 --> 0:12:49.840 you can still make everyone move over one room and 0:12:49.920 --> 0:12:53.040 thus open up a single room for me to stay 0:12:53.040 --> 0:12:56.360 in this evening. You, a whole person, a whole number, 0:12:56.720 --> 0:13:00.320 a natural person, a natural number, may and are into 0:13:00.320 --> 0:13:03.160 this hotel in fact, and if forty other people want 0:13:03.200 --> 0:13:05.880 to join you, they can, I mean not in your room. Obviously, 0:13:05.920 --> 0:13:07.480 you don't want to sleep with forty other people in 0:13:07.480 --> 0:13:10.160 your room. But forty other people would like to get 0:13:10.200 --> 0:13:13.760 forty rooms. Well, hey guess what they can. And all 0:13:13.800 --> 0:13:16.640 I have to do is have everybody gather their luggage 0:13:16.720 --> 0:13:20.280 and move to room N plus forty. So if you 0:13:20.320 --> 0:13:24.400 are in room two, now you move to room two. 0:13:25.880 --> 0:13:27.679 That sounds reasonable. I'm gonna pick up my bags, I'm 0:13:27.720 --> 0:13:29.520 gonna move on in and hopeful I'm gonna be a 0:13:29.520 --> 0:13:32.000 sleep in an hour or two. Oh. By the way, 0:13:32.120 --> 0:13:34.480 I see a bus coming around the corner. Looks like 0:13:34.520 --> 0:13:39.480 some more guests are arriving. Thus containing a countardly infinite 0:13:39.559 --> 0:13:43.240 number of people arriving. No problem. All I have to 0:13:43.280 --> 0:13:46.800 do is ask each guest to move to room N 0:13:47.160 --> 0:13:50.160 to room number two N. So room one moves room 0:13:50.160 --> 0:13:53.360 to room two to four Room three to six and 0:13:53.440 --> 0:13:55.920 four to eight, and this fills up all the infinite 0:13:56.000 --> 0:14:00.480 rooms and empties all the infinite odd rooms. So you're 0:14:00.520 --> 0:14:05.160 telling me that the Infinity Hotel, which is currently filled 0:14:05.240 --> 0:14:09.000 up with infinite number of guests, has room to take 0:14:09.040 --> 0:14:12.280 care of the guests the infinite number of guests arriving 0:14:12.360 --> 0:14:14.640 in an infinity bus. That's right, because all it is 0:14:14.640 --> 0:14:19.000 is shifted to everybody to the infinite even number rooms 0:14:19.400 --> 0:14:22.440 to accommodate all the infinite amount of people coming off 0:14:22.440 --> 0:14:25.080 the infinite buses. I'm gonna put them all in odd 0:14:25.120 --> 0:14:28.160 number rooms. And you remember Cantor in his infinite sets. 0:14:28.160 --> 0:14:30.480 This is an example of those one to one matches, 0:14:30.560 --> 0:14:34.200 not to mention, underscoring the idea that Cantor had that 0:14:34.320 --> 0:14:38.360 there are an infinite odd and even amount of numbers. 0:14:38.960 --> 0:14:42.320 So so, so far, the Infinity Hotel thought experiment is 0:14:42.360 --> 0:14:45.800 holding up. But of course, one of the the fun 0:14:45.840 --> 0:14:50.280 things about thought experiments is that you can continue to 0:14:50.320 --> 0:14:52.800 experience with them in an attempt to break them, to 0:14:52.920 --> 0:14:56.720 push them to the absolute limits. Break come on, Okay, Well, 0:14:57.240 --> 0:15:00.280 so far the Infinity Hotel has been able to deal 0:15:00.320 --> 0:15:02.960 with one new occupant. It's been able to deal with 0:15:03.480 --> 0:15:06.480 a bus full of infinite new occupants. But what does 0:15:06.520 --> 0:15:10.560 it do of coming around the corner? Are infinite buses 0:15:10.760 --> 0:15:16.320 and each bus has infinite yes, I mean just an 0:15:16.320 --> 0:15:22.520 infinite line of infinitely filled buses. All right, well, maybe 0:15:22.560 --> 0:15:25.560 I remember that youklid said that they're an infinite number 0:15:25.560 --> 0:15:29.080 of prime numbers. Yep, I just remembered that. And so 0:15:29.440 --> 0:15:33.200 current occupants of rooms are assigned the prime number two. 0:15:33.600 --> 0:15:36.160 So for instance, the current occupant of room number seven 0:15:36.160 --> 0:15:40.880 goes to two to the seventh power or room. Then 0:15:41.240 --> 0:15:44.040 the next group they're all assigned to the prime number three, 0:15:44.520 --> 0:15:48.280 and they take their bus seat numbers to figure out 0:15:48.360 --> 0:15:52.080 their room numbers. So seat number seven goes to three 0:15:52.080 --> 0:15:55.520 to the seventh power or room two thousand and seven. 0:15:56.040 --> 0:15:57.760 And I want to throw in it. This is important 0:15:57.800 --> 0:16:00.520 because if you just did them bust by bus, you 0:16:00.520 --> 0:16:03.920 would never finish onload owning the first bus. Well, yeah, 0:16:03.960 --> 0:16:06.360 first you gotta take uh, you gotta take care of 0:16:06.400 --> 0:16:08.880 your career that's already in in the hotel, right that 0:16:08.920 --> 0:16:10.800 you've given that prime number two, And then yes, you 0:16:10.840 --> 0:16:13.760 have to start referring to bus seat numbers to begin 0:16:14.680 --> 0:16:17.960 applying all the infinite number of prime numbers, so you 0:16:18.000 --> 0:16:20.480 can do this. Each each new group gets a new 0:16:20.560 --> 0:16:25.200 prime number, so you know, you go to the you know, 0:16:25.200 --> 0:16:28.640 prime number five, then prime number seven, eleven, and thirteen, 0:16:28.680 --> 0:16:31.360 and everybody just has to reference that number on their 0:16:31.440 --> 0:16:35.480 bus seat. So again, this is just really it's it's messy, 0:16:35.560 --> 0:16:40.320 but there's an elegant system to take place or to 0:16:40.400 --> 0:16:43.480 make sure that everybody gets their place, and that ensures 0:16:43.520 --> 0:16:46.360 that there are no overlapping room numbers for the infinite 0:16:46.360 --> 0:16:49.040 amount of people filing out of the infinite number of buses. 0:16:49.200 --> 0:16:52.480 Here's here's the thing, though, here's the thing. Okay, here's 0:16:52.480 --> 0:16:54.360 what I don't want to see coming out of those buses. 0:16:55.360 --> 0:16:59.240 I don't want to see any irrational numbers. I'm talking 0:16:59.240 --> 0:17:01.680 about you, pie. I don't want to see you tumbling 0:17:01.760 --> 0:17:04.280 off the bus because this is not going to work. 0:17:04.520 --> 0:17:08.160 I don't need to see a negative seven a representation 0:17:08.720 --> 0:17:13.040 human representation of a negative seven. First, you're you're negative, right, 0:17:13.119 --> 0:17:15.679 you're killing the vibe here in the hotel. Second, I 0:17:15.720 --> 0:17:21.000 don't have negative numbers. I don't have basement number rooms, 0:17:21.320 --> 0:17:24.880 you know, extending infinitely down into my my building here, 0:17:24.880 --> 0:17:29.359 and I don't have half fractional rooms available, because that 0:17:29.480 --> 0:17:33.120 ultimately breaks the hotel, that brings everything crashing down, because 0:17:33.160 --> 0:17:35.879 there's one infinity that's too big, and that's the infinity 0:17:35.920 --> 0:17:39.160 contained in a continuous flying, a continuous line of real 0:17:39.280 --> 0:17:41.720 numbers that goes on forever. If it showed up at 0:17:41.720 --> 0:17:43.959 the hotel, they wouldn't be room for everyone. You wouldn't 0:17:43.960 --> 0:17:46.119 be able to, uh to take that real number and 0:17:46.119 --> 0:17:47.320 count it. You wouldn't be able to put it on 0:17:47.320 --> 0:17:51.960 the guest list. Yeah, and that's yeah exactly, that's the problem. 0:17:51.960 --> 0:17:54.600 I mean, because we can deal with this lowest level 0:17:54.680 --> 0:17:59.120 of infinity, the countable infinity of the natural numbers, but 0:17:59.440 --> 0:18:04.720 that's uh, that real number, that continuum that Cantor had 0:18:04.760 --> 0:18:08.920 described before, that is uncountable, that doesn't behave in that 0:18:09.040 --> 0:18:12.680 one to one set way. And that's what I love 0:18:12.720 --> 0:18:17.600 about the Infinity Hotel is that it describes both these 0:18:17.760 --> 0:18:22.800 different levels of infinity, these larger amounts of infinity, but 0:18:22.920 --> 0:18:27.640 how certain numbers don't work within it or are uncountable. 0:18:27.800 --> 0:18:32.080 Plus you've seen being John Makovitch, right, you know what 0:18:32.240 --> 0:18:35.680 happens when you have a half demension, right, you had 0:18:35.760 --> 0:18:39.480 the half floor of the buildings right where where everything's 0:18:40.160 --> 0:18:42.560 cut in half and you have to stoop. And what happens, 0:18:42.960 --> 0:18:47.119 people will will use that as a vessel to somehow 0:18:47.359 --> 0:18:51.000 infiltrate your mind and hijack your body if you're placed 0:18:51.040 --> 0:18:53.600 in a half room or a half floor. Yeah, and 0:18:53.640 --> 0:18:55.919 that nobody wants that. It's gonna hard hard to retain 0:18:55.960 --> 0:18:58.560 tenants when you have that going on. Exactly, so again 0:18:58.600 --> 0:19:01.359 you have Cantor's continum hype offices. Fit says that there's 0:19:01.440 --> 0:19:04.560 there's no set whose cardinality is strictly between that of 0:19:04.640 --> 0:19:06.720 the integers and the real numbers. Can't prove it's true, 0:19:06.760 --> 0:19:09.199 can't prove it's false, can't measure it, can't hunt for 0:19:09.240 --> 0:19:12.240 a particle or measure anything about it. Yes, those real 0:19:12.320 --> 0:19:16.880 numbers are uncountable in this scenario of infinity. And then 0:19:16.920 --> 0:19:20.880 along uh of of and about nine gun named Goodill 0:19:22.000 --> 0:19:25.040 has a pair of proofs that are just spectacular to 0:19:25.080 --> 0:19:29.240 the math community because essentially he shows that you can 0:19:29.280 --> 0:19:33.520 never prove that the continuum hypothesis is false. And this 0:19:33.560 --> 0:19:37.320 feels like, uh, you're like you're moving the needle here 0:19:37.600 --> 0:19:39.960 on infinity just a bit right, Like we think we've 0:19:39.960 --> 0:19:43.440 grabbed onto something here. The problem is that in nineteen 0:19:43.600 --> 0:19:47.239 sixty or they're about Paul J. Cohen shows that you 0:19:47.240 --> 0:19:51.640 can never prove that the continuum hypothesis is true. And 0:19:52.119 --> 0:19:54.639 why does it all of this matter is because it 0:19:54.680 --> 0:19:59.679 shows that there are unanswerable questions in mathematics, particularly dealing 0:19:59.760 --> 0:20:03.560 with infinity, which just is really a microcosm of the 0:20:03.560 --> 0:20:10.080 macrocosm problem of the unknowability of life in general. So 0:20:10.119 --> 0:20:12.720 if your brain is exploding right now, don't worry, because 0:20:12.760 --> 0:20:15.040 now we're going to take it back to physics. We're 0:20:15.040 --> 0:20:19.080 gonna take it to discussions of the physical world as 0:20:19.160 --> 0:20:22.399 complicated and mind blowing is that whole realm can get 0:20:22.480 --> 0:20:26.200 at least we're dealing with observable physical reality, right, Sure, 0:20:26.240 --> 0:20:28.280 you say that, And now I have the physics sweats, 0:20:28.280 --> 0:20:31.680 which is an entirely different stench here. And that's understandable. Yes, 0:20:31.760 --> 0:20:36.160 but because certainly physics gets very complicated as well. But 0:20:36.280 --> 0:20:41.080 when but here's the thing. When we see infinities in physics, generally, 0:20:41.280 --> 0:20:43.159 it means that we have a problem and means that 0:20:43.400 --> 0:20:47.399 something's really catastrophic wrong unless you get into theoretical physics, 0:20:47.480 --> 0:20:49.840 as we will in a second. Yes, but you know 0:20:49.920 --> 0:20:53.560 basic physics, you're trying to use somebody that's designing. You know, 0:20:53.600 --> 0:20:56.240 you know a new building and you look at the plans. 0:20:56.240 --> 0:20:59.000 If you see infinity on there, you know that something 0:20:59.119 --> 0:21:02.000 is wrong. You do not want to try and stand 0:21:02.040 --> 0:21:04.959 in that building or or or or certainly have an 0:21:05.000 --> 0:21:08.639 office in it. Um it's never an actual measurement. It 0:21:08.680 --> 0:21:12.959 doesn't correspond to to to reality as as we we 0:21:13.040 --> 0:21:15.840 deal with it in our lives right today, for instance, 0:21:16.119 --> 0:21:20.040 quantum mechanics okay um and and and in such an 0:21:20.040 --> 0:21:23.800 important area of studies. So many of our greatest technological 0:21:23.800 --> 0:21:27.720 achievements in recent memory have arisen out of quantum mechanics. 0:21:27.720 --> 0:21:30.840 But even here we see this crisis. The crisis being 0:21:30.840 --> 0:21:33.920 that even though we can predict the the light heat 0:21:34.000 --> 0:21:37.600 power emission of a lamp, it's a finite amount of energy, 0:21:37.640 --> 0:21:43.560 but the various wigglings of waves and atoms creates this 0:21:43.640 --> 0:21:47.520 answer infinite energy, which which we simply can't deal with. 0:21:47.680 --> 0:21:50.239 Right we know that there's not infinite energy coming out 0:21:50.240 --> 0:21:53.199 of off of a lamp, so we afterwards we have 0:21:53.280 --> 0:21:56.240 to begin applying quantum physics to the forces of our universe. 0:21:56.280 --> 0:22:00.600 Quantum field theory, which is of course initially played infinities 0:22:00.640 --> 0:22:03.000 as well, and it takes ends up taking decades to 0:22:03.200 --> 0:22:05.520 flog through it all. You end up, you know, having 0:22:05.720 --> 0:22:08.480 all these infinities, all of these problems in the in 0:22:08.560 --> 0:22:12.359 the theory that have to be eradicated, like like enemies 0:22:12.359 --> 0:22:15.639 floating around in a video game, or which becomes incredibly 0:22:15.680 --> 0:22:19.119 important if you're looking at particle physics, right, because if 0:22:19.160 --> 0:22:23.040 you're trying to isolate the Higgs boson, you're taking these 0:22:23.200 --> 0:22:26.520 finite number of particles that we know and have named 0:22:26.600 --> 0:22:30.360 in seeing how they behave right, you do not need 0:22:30.440 --> 0:22:33.640 infinity in the mix if you're trying to figure out 0:22:33.680 --> 0:22:37.359 the composition of the universe, both from the time in 0:22:37.400 --> 0:22:40.760 which it became to now I mean, take the Higgs 0:22:40.760 --> 0:22:43.119 for example, the search for the Higgs boson, this was 0:22:43.680 --> 0:22:47.080 such the so called god particle, right. Uh. This uh, this, 0:22:47.200 --> 0:22:50.040 this of course been one of the predominant science stories 0:22:50.359 --> 0:22:54.280 in recent years, because the whole idea was that we 0:22:54.320 --> 0:22:56.800 had infinities in the theory, we had to remove them, 0:22:57.040 --> 0:23:00.480 and the thing that could remove it would be this 0:23:00.680 --> 0:23:03.919 Higgs particle. Right, So the whole quest for the Higgs 0:23:03.960 --> 0:23:08.200 particle comes out of this, this this necessity, this need 0:23:08.520 --> 0:23:14.280 to eradicate the infinities from the theory, because again, infinite's 0:23:14.359 --> 0:23:17.680