The Infinity Hotel

is not, because that's our nice little baseline there. It is not a real number, because we discussed before. You can't say that infinity is X number. You can't write it on a check. You can use a symbol, I guess, but good luck cashing it. Uh, why have I never done that? Uh? It cannot be measured. Of course, that is tied back to it not being a real number. It's not something that's growing, it's not doing anything. It's

the ultimate Zen concept. It just is yes. And of course this has been an important topic to philosophers, uh, theologians, great thinkers throughout time, and that's one of the primary things we discussed in our last episode because Ultimately you get into discussions about the boundless, about the limits of things. I mean, you're dealing with everything from the basic human experience. You know, how do I deal with the fact that

my life only stretches on until so so long? How do I deal with the fact that I can only perceive the universe with a certain bubble of perception. And then you start imagining a god and you're trying to figure out how that works. Is God is God? Infinite? Is God physically infinite? Is God infinite in quality as opposed to quantity. It's to say the biggest idea is it is the the the infinite idea that the humanity

has tackled with over the ages. Yeah, and we talked about George Tvorski writing for Ion nine, and he used the chess board to illustrate this idea, saying that all the other pieces having number assigned to it except for

the king, whose number value is infinite. Yeah, that chess is It is interesting because we mentioned before, you have a finite number of playable games for chess, and then you then you have this king that has an infinite importance within the game, So you have two different types of infinity going on there. Yeah, you lose, you lose the king, You lose the game because zer or some game if you get rid of the king, and so again it ties back to this whole idea about eternity

even consciousness uh embedded in there with infinity. So of course it becomes with this great uh media area for philosophers to really tussle with, because it's got the afterlife there, it's got the idea and the question of God, and by its very nature, it's an idea that we cannot fully contain and we cannot really completely grasp it, and yet we try to and so did philosophers as we discussed the last episode, and they used to try to prove this out or try to make their arguments even

more persuasive. So today we're gonna look at some examples in math and physics to see how well we can contain the idea of infinity and reach some sort of meaning, which we already talked about as sort of impossible, like meaning is infinite in itself for various reasons. To just to recap mathematics and infinity, you voluencountered numbers like twelve point one to nine, eight, seven, six, five, three, two, one, zero, one, just on on and on forever. You just have to

stop counting those numbers. After a while, you have to stop recording them because it becomes a pointless exercise. You just round up, right. Is that because humans are finite and we kind of continue to count infinitely. Yeah, that's that's Uh, that's kind of the big topic of discussion here. And so you kind of have two schools of thought in mathematics, right when when it comes to to struggling with the infinite, and it kind of comes down to

some of the core arguments with mathematics itself. So is is mathematical infinity? Is this a mere human creation, an extrapolation into the world where there's no truth? Or does mathematical infinity actually exist? Well, this comes down to the question of God to rite. There are some observable things of how infinity works in mathematics, um, and I say observable, but you can it's just as we discussed in your example. There numbers can go on and on forever, and we

have finite lives. Uh, we can't see God in Therefore, some people would say he or she or it does not exist. So you're grappling with some of the same concepts in math and in infinity. So if you were to ask the constructivists or the institutionists, they would say that classical mathematics, well, they deal with the sort of math that God would do. An infinite God he can tackle with infinite numbers. Uh, and and and the idea

of a mathematical infinity. But we are humans, we are finite, and we would rather focus with human mathematics with finite numbers because that ultimately is what relates to what we do. You know what I hear with that, my brain hurts. I'll just I'll just leave it to God. He or she or it can deal with it. Yeah, I mean, it's kind of the idea that infinity is an interesting topic to tackle with as an abstraction, but does it really relate to the work that I'm doing here, the

work that mathematics needs to do in the world. And then they're the formalist Now. Formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain certain string manipulation rules. Um systematic formulation of the concept of mathematical formulism arose directly as a reaction to the paradox discovered

within set theory, which studies the concept of infinity. According to twenty century mathematician Abraham Robinson, infinite totalities do not exist, either really or ideally. Any mentioned of infinite totalities is meaningless. Still, the business of mathematics must continue as if infinite totalities actually do exist. Right, we said, there are a couple of things that we're talking about. A system here in

this idea that you can again capture infinity. You can define it um and then set theory, which is something that a man named Georg Cantor came up with. And

he's sort of the rabble rouser here. According to Natalie while cover writing for Quantum magazine, infinity was boxed and sold to the mathematical community in the late night concentury by the German mathematician Georg Cantor, who invented a branch of mathematics dealing with sets collections of elements that ranged from empty the equivalent of the number zero to infinite. So Cantra stepped in there really and began to define

infinity in a much more specific way. Yeah, and uh he was he was a one of these individuals who was hated as well as love, depending on who you're asking, because he was bringing up some really mind blowing, disturbing takes on infinity. Yeah. Natalie Walquiver says that other mathematicians initially despise what they called this mess of infinities. And we'll talk more about this because it's largely due to his ideas of set theory, and one of his colleagues

called them a grave disease. Another called him a corrupter of you. Uh. Sadly, he was so vilified that he actually um fill into a lifelong depression after that. Um. But I think it tells you how radical some of these ideas were at the time and still are really, because what it does is it just sets minds ablaze as to what what reality is and isn't. Essentially, Yeah, I mean, one of the core concepts to come out of it is that there are different kinds of infinity

and some are bigger than others. Well, that that's thinking for a second. Yeah, let me let's get into the math sweats here. I'm having the math sweats right now. Uh. Set theory is essentially a useful language for describing mathematical objects. And what we're talking about is a nine item list of rules that Cantor came up with called Zermelo Frankel set theory with the axiom of choice or the z f C, established and widely adopted in math by the

nineteen twenties. Okay Um. One of the axioms says that two sets are equal if they contain the same elements. Another simply asserts that infinite sets exist. So you can have these infinite sets that are one to one equal to each other, and you have them going on forever and ever and ever and ever. And Cantor showed that for any infinite set, forming a new set made of all the subsets of the original sets represents a bigger

infinity than that original set. So once you have one infinity, you can always make a bigger one by creating a subset the original set, and then an even bigger one by making a set of all the subsets, and so on and so forth. And there are an infinite amount of infinities in different sizes. And he also proved doubt that an infinite set of even numbers like two, four, or six could be put in that one to one

correspondence with all counting numbers like one to three. So that came to this idea that there are as many evens as there are odds and evens in an infinite set. There's one problem with this, though, real numbers, yes, like the one that you talked about earlier. You know that for example point zero zero zero one or pie right three point one four or so on and so forth, blah blah blah blah. They go on for an ever and they are uncountable, and they don't correspond in a

one to one fashion with counting numbers. But hold on because we will get back to that. If that didn't thinking completely. Will return to the unccountability of real numbers in a minute, once we have a structure in which to house them. All right, I am going to dab at the infinite number of sweat molecules dripping off of me right now. Uh No, it's not too bad actually, Uh And let's take a break. When we get back,

you'll talk more about the infinite hotel. If you're out there and you're thinking, I don't know what they're talking about there, they're losing me. I'm sinking beneath the waters here in the It is the in the deep end of the mathematical, philosophical, theological, physical pool. I need help, while we need help to for the love of infinity, give me an example. Yes, we do have, We do

have an example. We have a flotation device. And it is a thought experiment, because the thought of experiments, as always they can take some very difficult to grasp ideas and concepts and put them into into a metaphorical form that we can latch onto and then and then actually explore the concept a little more easily and sometimes a little a little deeper. Right, And what could be more of a form inaccessible than a building, which is essentially

what we're talking about. Um, this is an infinite building, an infinite hotel. And this was created by German mathematician David Hilbert who was obsessed with cantors work and and came up with us little ditty to try to explain a hotel with an infinite number of rooms. All right, it's so it's a hotel, infinite number of rooms? How many how many rooms? Does it have to like an infinite amount? Oh but accountable infinite amount, which is going

to turn out to be important. All right, So let's say just to just to roll out the basic entry level portion of the thought experiment. I show up to the Infinity Hotel, all right, and and I need a room for the night. I know there's a convention in town. Uh, the local set theory convention maybe, and uh, and I I know that all the hotels are booked, but I really need a room. So the Infinity Hotel sounds like a good place to go, all right, So let me

a my jaunty little manager's bellhop cap here. What I am going to do is, I'm going to ask the guests in room do you have vacancy? Always? We have an infinite amount of rooms. Okay, I see a lot of cars in the parking lot. Hold up, okay, just look at a magazine or something. We'll figure this out. I'm going to ask the guests in room one to move to room two, and then the guests in room two to move to room three, and so on and so forth. Every guest moves to room N plus one.

And since there are an infinite amount of rooms here, then there's room for every guest to move into a different room. So you're saying that even though they're already infinite guests at the infinity hotel. Occupying the infinite rooms, you can still make everyone move over one room and thus open up a single room for me to stay in this evening. You, a whole person, a whole number, a natural person, a natural number, may and are into this hotel in fact, and if forty other people want

to join you, they can, I mean not in your room. Obviously, you don't want to sleep with forty other people in your room. But forty other people would like to get forty rooms. Well, hey guess what they can. And all I have to do is have everybody gather their luggage and move to room N plus forty. So if you are in room two, now you move to room two. That sounds reasonable. I'm gonna pick up my bags, I'm gonna move on in and hopeful I'm gonna be a sleep in an hour or two. Oh. By the way,

I see a bus coming around the corner. Looks like some more guests are arriving. Thus containing a countardly infinite number of people arriving. No problem. All I have to do is ask each guest to move to room N to room number two N. So room one moves room to room two to four Room three to six and four to eight, and this fills up all the infinite

rooms and empties all the infinite odd rooms. So you're telling me that the Infinity Hotel, which is currently filled up with infinite number of guests, has room to take care of the guests the infinite number of guests arriving in an infinity bus. That's right, because all it is is shifted to everybody to the infinite even number rooms to accommodate all the infinite amount of people coming off the infinite buses. I'm gonna put them all in odd

number rooms. And you remember Cantor in his infinite sets. This is an example of those one to one matches, not to mention, underscoring the idea that Cantor had that there are an infinite odd and even amount of numbers. So so, so far, the Infinity Hotel thought experiment is holding up. But of course, one of the the fun things about thought experiments is that you can continue to experience with them in an attempt to break them, to

push them to the absolute limits. Break come on, Okay, Well, so far the Infinity Hotel has been able to deal with one new occupant. It's been able to deal with a bus full of infinite new occupants. But what does it do of coming around the corner? Are infinite buses and each bus has infinite yes, I mean just an infinite line of infinitely filled buses. All right, well, maybe I remember that youklid said that they're an infinite number

of prime numbers. Yep, I just remembered that. And so current occupants of rooms are assigned the prime number two. So for instance, the current occupant of room number seven goes to two to the seventh power or room. Then the next group they're all assigned to the prime number three, and they take their bus seat numbers to figure out their room numbers. So seat number seven goes to three to the seventh power or room two thousand and seven.

And I want to throw in it. This is important because if you just did them bust by bus, you would never finish onload owning the first bus. Well, yeah, first you gotta take uh, you gotta take care of your career that's already in in the hotel, right that you've given that prime number two, And then yes, you have to start referring to bus seat numbers to begin applying all the infinite number of prime numbers, so you

can do this. Each each new group gets a new prime number, so you know, you go to the you know, prime number five, then prime number seven, eleven, and thirteen, and everybody just has to reference that number on their

bus seat. So again, this is just really it's it's messy, but there's an elegant system to take place or to make sure that everybody gets their place, and that ensures that there are no overlapping room numbers for the infinite amount of people filing out of the infinite number of buses. Here's here's the thing, though, here's the thing. Okay, here's what I don't want to see coming out of those buses. I don't want to see any irrational numbers. I'm talking

about you, pie. I don't want to see you tumbling off the bus because this is not going to work. I don't need to see a negative seven a representation human representation of a negative seven. First, you're you're negative, right, you're killing the vibe here in the hotel. Second, I

don't have negative numbers. I don't have basement number rooms, you know, extending infinitely down into my my building here, and I don't have half fractional rooms available, because that ultimately breaks the hotel, that brings everything crashing down, because there's one infinity that's too big, and that's the infinity contained in a continuous flying, a continuous line of real numbers that goes on forever. If it showed up at

the hotel, they wouldn't be room for everyone. You wouldn't be able to, uh to take that real number and count it. You wouldn't be able to put it on the guest list. Yeah, and that's yeah exactly, that's the problem. I mean, because we can deal with this lowest level of infinity, the countable infinity of the natural numbers, but that's uh, that real number, that continuum that Cantor had described before, that is uncountable, that doesn't behave in that

one to one set way. And that's what I love about the Infinity Hotel is that it describes both these different levels of infinity, these larger amounts of infinity, but how certain numbers don't work within it or are uncountable. Plus you've seen being John Makovitch, right, you know what happens when you have a half demension, right, you had the half floor of the buildings right where where everything's

cut in half and you have to stoop. And what happens, people will will use that as a vessel to somehow infiltrate your mind and hijack your body if you're placed in a half room or a half floor. Yeah, and that nobody wants that. It's gonna hard hard to retain tenants when you have that going on. Exactly, so again you have Cantor's continum hype offices. Fit says that there's there's no set whose cardinality is strictly between that of

the integers and the real numbers. Can't prove it's true, can't prove it's false, can't measure it, can't hunt for a particle or measure anything about it. Yes, those real numbers are uncountable in this scenario of infinity. And then along uh of of and about nine gun named Goodill has a pair of proofs that are just spectacular to the math community because essentially he shows that you can

never prove that the continuum hypothesis is false. And this feels like, uh, you're like you're moving the needle here on infinity just a bit right, Like we think we've grabbed onto something here. The problem is that in nineteen sixty or they're about Paul J. Cohen shows that you

can never prove that the continuum hypothesis is true. And why does it all of this matter is because it shows that there are unanswerable questions in mathematics, particularly dealing with infinity, which just is really a microcosm of the macrocosm problem of the unknowability of life in general. So if your brain is exploding right now, don't worry, because

now we're going to take it back to physics. We're gonna take it to discussions of the physical world as complicated and mind blowing is that whole realm can get at least we're dealing with observable physical reality, right, Sure, you say that, And now I have the physics sweats, which is an entirely different stench here. And that's understandable. Yes, but because certainly physics gets very complicated as well. But

when but here's the thing. When we see infinities in physics, generally, it means that we have a problem and means that something's really catastrophic wrong unless you get into theoretical physics, as we will in a second. Yes, but you know basic physics, you're trying to use somebody that's designing. You know, you know a new building and you look at the plans. If you see infinity on there, you know that something

is wrong. You do not want to try and stand in that building or or or or certainly have an office in it. Um it's never an actual measurement. It doesn't correspond to to to reality as as we we deal with it in our lives right today, for instance, quantum mechanics okay um and and and in such an important area of studies. So many of our greatest technological achievements in recent memory have arisen out of quantum mechanics.

But even here we see this crisis. The crisis being that even though we can predict the the light heat power emission of a lamp, it's a finite amount of energy, but the various wigglings of waves and atoms creates this answer infinite energy, which which we simply can't deal with. Right we know that there's not infinite energy coming out of off of a lamp, so we afterwards we have to begin applying quantum physics to the forces of our universe.

Quantum field theory, which is of course initially played infinities as well, and it takes ends up taking decades to

flog through it all. You end up, you know, having all these infinities, all of these problems in the in the theory that have to be eradicated, like like enemies floating around in a video game, or which becomes incredibly important if you're looking at particle physics, right, because if you're trying to isolate the Higgs boson, you're taking these finite number of particles that we know and have named in seeing how they behave right, you do not need

infinity in the mix if you're trying to figure out the composition of the universe, both from the time in which it became to now I mean, take the Higgs for example, the search for the Higgs boson, this was such the so called god particle, right. Uh. This uh, this, this of course been one of the predominant science stories in recent years, because the whole idea was that we had infinities in the theory, we had to remove them, and the thing that could remove it would be this

Higgs particle. Right, So the whole quest for the Higgs particle comes out of this, this this necessity, this need to eradicate the infinities from the theory, because again, infinite's just don't work in the set and the infinities here in the theory said said, all right, we have an infinity. That means there's something finite in the world that we haven't discovered yet, and if we discover it, then we

find our way around that answer. Now, if you sort of back up and then begin to take the large view of the universe and not just try to contain it in this this one model, right, you know that the universe is expanding how much infinitely? These are questions that come up, and then you begin to say, well,

how how far does space go? Anyway? And this is where theoretical physicists get into infinity and they really have some fun um And it's fascinating because here you see some examples of infinity played out in ways that you say, this is a possibility, because infinity otherwise we just see is this sort of you know, the line on the piece of paper and mathematics that goes on forever. But

then you've got these ideas like multiverse existing. Now, when I think of the multiverse, I always think of Jorey Lewis Borges the Library of Battle, the library that first of all contains all books, all written books, but then beyond that also contains all possible books, and and and and and so being is is ultimately kind of a model for a multiverse. Uh, an existence that contains everything

that is and everything that could be. Again, you go back to Cantor, and you're talking about larger infinities, right, And essentially that's what we're talking about with multiverses. And I'm going to read this from Brian Green's The Elegant Universe. Um, we could talk about multiverse in its very own episodes.

We won't go too deep here, he says, Imagine that what we call the universe is actually only one tiny part of a vastly larger cosmological expanse, one of the nor almost number of island universes scattered across a grand cosmological archipelago. Although this might sound rather far fetched, and in the end it may well be, Andrei Lynd has suggested a concrete mechanism that might lead to such a

Gargangan universe. Lynn has found that the brief but crucial burst of inflationary expanse may not have been a unique one time event. Instead, he argues the conditions for inflationary expansion may happen repeatedly in isolated regions and the peppered throughout the cosmos, which then undergo their own inflationary ballooning

and size, evolving into new separate universes. And each of the universes, the process continues, with new universes sprouting from far flung regions in the old, generating a never ending web of ballooning cosmic expanses. And this is what he calls the multiverse. And that's pretty mind blow I mean, even if you you shrink back down to are single universe. Um. There was an interesting point that was brought up by physicist Raphael Bosso in that World Science Festival talk I

was mentioning earlier. We will include a link to in the landing page for this podcast episode. But he pointed out that that when you start start looking at the way light travels across the universe, when you start looking at this expansion of the universe, you end up with a universe that's arbitrarily large. Light can never reach you from rapidly expanding regions regions, and so any given observer is trapped within a finite sphere of observable universe within

an expanding infinite. So you kind of get this again. You get these ideas of here's this, here's the sphere of the finite within the infinite. You can think about the infinite, but you're ultimately trapped within that sphere of the finite. Yeah, and this is a similar idea that Lee Smollen has in terms of the perception of the

infinite at least. Smallen, by the way, is a cosmologist at and State, and his idea is that the conditions at the Big Bang and at the centers of black holes, each being characterized by a colossal density of crushed matter. I suggested that every black hole is the seed for a new universe that erupts into existence through a Big Bang like explosion, but is forever hidden from our view

by the black holes event horizon. And of course, any of these discussions of the physical universe are are even more twisted when you when you have coorse draw in the fact that time and space are one, and if you play with space, you're playing with time, and if it were to be flat, then it could stretch out for an infinity, right, And then you get all these different ideas of well maybe in this case that supports this idea of some sort of repeating going on because

you have a finite number of particles that you have infinite space and time, and maybe that repeating pattern creates more universes, right, And then you have ever many worlds interpretation that says that the universe branches off into distinct worlds to accommodate every single possible outcome. And so maybe we live in an infinite web of alternate timelines. I mean, it gets crazy and crazy as you go along. So yes, I mean I don't even have to everyone else's mind

is extrapolating the possibilities on that. I don't have to push you in that direction. And if you guys would like some articles to accompany this, uh, we have ever many worlds interpretation on how stuff works, and we also have some good amount of string theory and multiverse um, so we definitely have some stuff for you guys to dive into. Yes, on the Internet, which is of course a finite world. Even though it is it is a finite world. But yet we we can't even grasp what

We can't even put a number on it. We can't say how many pages there actually are on the Internet. We can't actually even comprehend it's a finite thing we've created, and it is already, at least as far as human perception goes, bordering the infinite. Let's supposed to just fine, since we can see its growth and we can put a number on it. But it feels infinite at times. Alright, So we're gonna walk you guys out of this um and we're gonna take a little walk into something called

the infinite monkey theorem to end this section. Yeah good. This is an you know, a thought experiment that I often forget even really entails infinity, maybe because I just get too caught up on the idea of monkeys banging on typewriters and now awful that writing room, let's be, and I've I've written in some awful rooms before. Uh. This states that a monkey hidden keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, like say, the complete

works of William Shakespeare. Okay, okay, And so in this context, almost surely is a mathematical term, alright, and it has a precise meaning, And that monkey is not an actual monkey but a metaphor for an abstract device that produces a random sequence of letters for an infinity. In the theorem illustrates the perils of reasoning about infinity by imagining

this vast that finite number, and vice versa. So the probability is really tiny, right, The probability exists that a monkey could eventually write a work as cohesive as say Shakespeare's Hamlet, just by pure dumb accident of banging on the keys forever and ever and ever endever, which gives my fiction writing some hope. Oh you're you're better than

a monkey. I don't know. I don't know about that, and I'm fine with that actually, all right, So if you're still with us, then hopefully you have you maybe you have a better idea of what infinity is all about. Maybe you have more nuanced idea, more expansive idea. Maybe this is uh forced you to sort rearrange your your contemplation of the infinite and of the boundless, in terms of of our human experience, in terms of our cosmos, in terms of our ideas of God. So we'd love

to hear from you. We'd love to hear your thoughts about infinity, your personal takes on infinity, your favorite uses of infinity and fiction. All of that is fair game. Yeah, and before you do that, make sure you stopped at Stuff to Blow your Mind dot com. That's right, That's where you'll find all the podcast episodes, all the videos, all of the blog posts, a finite amount of all of those, but certainly plenty of stuff to keep you occupied. Yeah,

So share those thoughts with us, why don't you? And you can do that at below the mind house to works dot com. For more on this and thousands of other topics, visit how stuff works dot com