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55: Order in the Court (Transfinite Ordinal Numbers)
464 Plays4 years ago
As a child, did you ever have a conversation that went as follows:
"When I grow up, I want to have a million cats"
"Well I'm gonna have a billion billion cats"
"Oh yeah? I'm gonna have infinity cats"
"Then I'm gonna have infinity plus one cats"
"That's nothing. I'm gonna have infinity infinity cats"
"I'm gonna have infinity infinity infinity infinity *gasp* infinity so many infinities that there are infinity infinities plus one cats"
What if I told you that you were dabbling in the transfinite ordinal numbers? So what are ordinal numbers? What does "transfinite" mean? And what does it mean to have a number one larger than another infinite number?
[Featuring: Sofía Baca; Diane Baca]
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This episode features the song "Buffering" by "Quiet Music for Tiny Robots"
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Transcript
Introduction to Infinity and Ordinals
00:00:00
Speaker
As a child, did you ever have a conversation that went as follows? When I grow up, I want to have a million cats. Well, I'm going to have a billion, billion cats. Oh, yeah? Well, I'm going to have infinity cats. Then I'm going to have infinity plus one cats. That's nothing. I'm going to have infinity, infinity cats. I'm going to have infinity, infinity, infinity, infinity, infinity, infinity, so many infinities that their infinity, infinity is infinity plus one cats.
00:00:30
Speaker
What if I told you that you were dabbling in the transfinite ordinal numbers? So what are ordinal numbers? What does transfinite mean? And what does it mean to have a number that is one larger than another infinite number? All this and more on this episode of Breaking Math. Episode 55, Order in the Court.
00:00:53
Speaker
I'm Sophia, and this is Breaking Math.
Guest Introduction: Diane Baca
00:00:56
Speaker
With us we have on Diane Baca, who is an educator in the Albuquerque Public School systems, teaches math, and also happens to be my mom. And I did not introduce her proper last time. I was just like, this is my mom. Welcome. Thank you. I'm happy to be here again.
Support and Social Media
00:01:10
Speaker
So before we start on this episode I have a couple of plugs if you want to buy our poster Facebook.com slash breaking mouth podcast is where you can buy that I just click on shop the posters $15 and 15 cents plus $4 and 50 shipping handling
00:01:25
Speaker
coming to a total of $19.65, anywhere in the United States. If you want to donate to the show regularly, we have a Patreon at patreon.com slash breakingmath. If you donate a dollar or more, you can have episodes without ads. And we also even have outlines that we use for the show. If you want to spend $22.46 there every month, you could also get the poster, but that'll be more, you could get a cheaper on Facebook, but where's your loyalty there?
00:01:49
Speaker
If you could also find updates about the show on Facebook, we also have a Twitter at Breaking Math Pod, a website breakingmathpodcast.app, and a sister podcast touring rabbit holes.
Ordinal vs Cardinal Numbers
00:01:59
Speaker
So this episode is about the trans finite ordinal numbers. What do you think about that?
00:02:05
Speaker
I know the only word in there is I know, I know lots of like parts of that number, but I know ordinal, put them in order. Transfinite means what? Across infinite or across finite? Yeah, across the finite number is basically an infinite number. And there's actually a difference between ordinal numbers and cardinal numbers when you get into the infinities, which we're going to talk about cardinal numbers on the next episode. This is part one, basically.
00:02:31
Speaker
But, um, ordinal numbers are, let's say, I mean, let's say you have like a collection of things, let's say you have an infinite collection of things. Let's just say you have a million names, like an infinite number of names on a list and, uh, you have them all numbered from, uh, one, uh, one onward. And they're unique, right?
00:02:48
Speaker
Yeah, and they're all unique, or at least they have a unique address. If you add, let's say you add one to that, like one new name to that. While you could reorder them and say that they are... Well, is that one that you're adding? It's one more than the one before.
00:03:04
Speaker
Yeah,
Infinity Hotel Paradox
00:03:05
Speaker
and it's distinct. And you could order them basically the same way because that could be the new number one, the old number one could be number two, the old number two could be three and so forth. But if you want to compare, if this item is distinct, you can refer to it as kind of like the infinity plus one address. It's when you have a collection that's already infinite and you're adding one to it.
00:03:27
Speaker
So that makes no sense to me. It makes no sense to me because infinity to me means without end. Right. And so it seems to me like it would be immaterial to add another number in there because it's already there anyway. Like in other words, it's just basically that is when you're talking about cardinal numbers. And I'll explain the difference real quick. Cardinal numbers are how many there are. Ordinals are where they are.
00:03:52
Speaker
Oh, so we're not talking about. So when you were talking about names, we're literally not talking about ordering the numbers. We're literally talking about unique things. Yeah. Like, let's say like, um, you have Jimmy Bob and he's number one and then you have Muhammad. He's number two. I got it. And they have Pluto and summer three. Got you. Okay. I was thinking this was like one, two, three, four, five.
00:04:12
Speaker
Oh, yeah. I mean, one, two, three, four and five are ordinal numbers and they're also cardinal numbers because you can have a set, you know, like a collection of things with five things in it. But once you have an infinite number of things, like, for example, let me use the example before where we have the infinite names and then we add one to it. Right. Right. Okay. The address of the new thing can be considered like the infinity of plus one address.
00:04:31
Speaker
but the total number of things in it is still just the same infinity because cardinal numbers is about the size of sets and ordinal numbers are about addressing sets giving each place everything a unique name. So for in the example before we have everything numbered one through
00:04:53
Speaker
infinity. And then we add another item to it. And then if we compare that list, the old list to the new list, we could talk about this in terms of ordinal numbers.
00:05:03
Speaker
Okay, but they're still both infinity. Yeah, they haven't they have an infinite number of things in them However, if you want to if you if everything already has a unique address The only other unique address you could put tack on is infinity plus one So what doesn't make I guess maybe why I'm struggling with infinite the concept of infinite even if you had infinite unique I mean to me if you had infinite unique addresses Then that would mean that there is no end and there is no
00:05:33
Speaker
Size of set you couldn't talk about the size of the set because it is infinity it goes on forever and ever and ever and ever so You see what I'm like. Maybe my brain is more concrete to talk about this. Maybe we should talk about The infinity hotel first and then we'll talk about Omega because there's a concept we have to cover
00:05:52
Speaker
infinity hotel.
Prime Numbers in Infinity Hotel
00:05:54
Speaker
I feel like you did this when you were like 10 or 11 and you tried to get me to engage in this concept and I remember being pretty resistant because I just wanted to know what I was cooking for dinner. Anyway. I remember being at McDonald's and writing a stack of infinities on top of each other but I was wrong because that is not rigorous and we'll dive right in. Okay let's do it.
00:06:18
Speaker
Let's say that somebody has a hotel. Let's call it the infinity hotel and has a lot of rooms in it. It has an infinite number of rooms. Okay. Now let's say people show up, right? You could put person one in the first room, the second person in the second room, and then so on. Okay. Now let's say you did this forever. Okay. Essentially. And all the rooms are quote unquote full, right? Yeah. Now let's say, wait, there's empty rooms, aren't there?
00:06:45
Speaker
No, we're assuming that we filled it all up. It took an infinite number of time. So it seems like in this case, it almost sounds like infinite is finite because as if there's an end to it. Actually, yeah. What's cool about the trans-finite ordinals is that you can use them to prove really cool properties about finite numbers. It's kind of weird. We're going to talk about bases
Omega and Infinite Processes
00:07:08
Speaker
in a sec, but I'm sure you're loving this episode.
00:07:13
Speaker
Actually, it's okay. It pushes me. It's good. I like it.
00:07:16
Speaker
So your question about, like it is valid question, like how can an infinite number of rooms get filled up? And we have to kind of go to the concept of limits in a way for that. So, you know, like a limit in calculus, it's like as X approaches infinity, one divided by X approaches zero. And what that basically means obviously is that if you keep increasing the value of X, there's gonna be a number that it approaches.
00:07:45
Speaker
We could pick that number out without having to go through an infinite number of steps, but we theoretically have gone an infinite number of steps. Okay. Yes. Yes. Yes. I get that. Okay. So let's, so let's introduce the concept of Omega to talk a little bit about the infinity people. Okay. Now what Omega is, is let's say we start at one, right? I mean, you start at zero, whatever. Zero, one, two, three, four, and so on, right? Mm-hmm.
00:08:12
Speaker
The idea of counting just because remember with the limits, it's a real number that we're talking about. So we could go to like 2.5 to 5 to 7.5 on our way to infinity. But here we're just going 0, 1, 2, 3 all the way to infinity.
00:08:28
Speaker
Okay. So the basically the term and so on can be used to go from one level of infinity to the next. So let me give an example with the number of people. So let's say the hotel rooms, the hotel rooms are numbered, right? You're numbered one, two, three, and so on, right?
00:08:51
Speaker
Basically, the address of the quote-unquote last person there would be Omega. And of course, there is no actual Omega, but Omega is this concept of speeding along counting for an infinite amount of time. And you can imagine this after taking an infinite amount of time, and that's why maybe the infinity hotel is cool to mathematicians and can infuriate some other people.
00:09:15
Speaker
But what's cool about it too is that it also describes how these infinities have the same cardinality, and I'll explain. So let's say a bus full of infinity people arrive at the hotel, right? Okay, got it. How do you give them new rooms? What do you mean? We already got Omega rooms. Yeah, and they're all full. So you have to add something to that Omega.
00:09:40
Speaker
Well, yeah, but we can't add new rooms. The hotel is done, but the contractors are gone. They're mad at us. The one infinity is as much as we can do. Well, it's not as much as we can do, but as much as as many as we have. We're working with the hotel that it's in the historic register. You can't alter it. OK, I can live with that. All right. So what do we do? Yeah. So you have like, you know, Jim Bobbin room one, you have Mohammed in room two. You have do we have to put two people in a room now?
00:10:10
Speaker
Um, no, actually, no, everybody has to have their own room. Oh, and here's one of the rules. Uh, you can move people from room to room. I can move people to room to room. So how about I, but I don't have enough. I don't have any more rooms because otherwise I was going to say, Hey, just move them down the line. And then you add the new people in, but that actually, let's go with, I forgot that, um, the first example here is not a bus of infinity people, but a bus of like 30 people. Well, that's easy. You just put them in the 30 rooms. Yeah.
00:10:36
Speaker
Yeah, you have everybody move up 30 rooms. Oh, you're talking about the we've already filled we're already at the Omega and now we've got 30 more people Yeah, here's the thing. Here's the weird part too because the amount of rooms is infinite. It's We can actually have everybody including the last person move down the quote-unquote last person who doesn't exist move down 30 rows So we make 30 we make room for 30 in the front
00:11:00
Speaker
Yeah, but let's say the and every person that comes along we just we're the worst hotel in the world because we keep telling everybody to move rooms, but Yeah, every time somebody arrives everybody moves to the next room. That would suck. Okay, it'll be it would be like that scene in Alice in Wonderland Move down move down move down
00:11:24
Speaker
But so a bus now comes with infinity people because they're from an infinite festival. Okay. So now you have infinity plus infinity. How do you move everybody down an infinite number of rooms?
00:11:39
Speaker
I don't know, you move them Omega, I don't know. So you can't actually move them Omega because Omega is about an address. And you can't really move everybody an infinite amount because of rooms because you have to give everybody a direct address to go to. So you can't tell person one to go to room infinity plus one because that room in a way doesn't exist. Okay, but didn't we just do that when we added the 30 people?
00:12:05
Speaker
we added we added 30 people but they had the same cardinality because it's the same number of people because if you already have an infinite number of people it's the same thing as the it's the same concept as the why there's the same number of numbers and even numbers there's not more or less even numbers than numbers and you might think that there'd be half of them
00:12:27
Speaker
Okay. My mom is doing an explosion of emotion with her head. There's no more numbers than even numbers. Okay. Yeah. So like one, two, three, four, five. There's, there's the same amount of numbers like that as there are even numbers or numbers that are a multiple of 10 simply because you can keep going up. Yeah. As you count, you count them to infinity. Okay. So this is definitely a mind. Ooh. Yeah. But, um, that's my, my definition of something that's hard to understand online. Okay.
00:12:52
Speaker
But yeah, it's, oh yeah, but what I was gonna say too is the reason why the even numbers are like that is because if you count the even numbers, the first even number is two, the second even number is four, and you keep counting, and you keep counting to infinity, so that numbers are, the infinities are equal.
00:13:10
Speaker
However, there's way more, but weirdly enough, there's the same amount of fractions as there are numbers. Because the fractions, you could order them. You could do one, then one half,
Goodstein's Theorem and Piano Arithmetic
00:13:19
Speaker
then one third, then two thirds, and then one quarter, then three quarters, and just keep going like that. And you could write out every fraction like this. But back to the infinity hotel. So a bus of infinity people arrive. How do you assign them all a room? If you have a loudspeaker, you don't even have to do that. But how would you do this?
00:13:39
Speaker
I have no solution because if we can't move them down and if Omega is not really something we can do like Omega plus infinity or whatever it is. Well we can move people down as long as we know where we're moving them to. Well we're at we're approaching Omega correct Omega infinity or whatever. Here's a hint what I said about the same number of even numbers as natural numbers applies to this.
00:14:02
Speaker
So it's the same number of rooms? Yeah, the number of rooms doesn't ever change. So we have enough room for them? Oh yeah, we definitely have enough room for them.
00:14:10
Speaker
So now we don't, we don't have, so even though we have somebody in every room now, suddenly we have more rooms. Let me give you the solution and then maybe the next one will be... Please, what's the solution? What you do is you get on your loudspeaker and you tell everybody, all right everybody, double your room number and that's your new room. And then you assign everybody who comes in an odd room number. What? That feels like a cheater way. Infinity is full of this.
00:14:39
Speaker
Say you haven't just changed their room number. Yeah, because that's like moving them down. Okay. So now let's say. So you basically just created new rooms. I mean, you didn't, but you kind of did. Well, we didn't, we didn't create any new rooms. We just had, we just found, we just found a way to weave the people we already have into the rooms that we have.
00:14:59
Speaker
Sounds like infinity times two. Okay, anyway. And actually, yeah, because if, let's say that you have like the Omega people in the hotel, if we could say that the set of the people in the hotel, like combined with the set in the buses is two times Omega.
00:15:18
Speaker
Okay, but once they're all renumbered together, it's just Omega again. Yeah, it's just Omega again. Okay So this these are weird numbers to use. So now let's say that it's a festival, right? It's a really turned out to be a really big festival So now here come infinity buses each with infinity people. Okay So what are we gonna multiply the room number by infinity or something? Oh
00:15:43
Speaker
Actually, no in this this one it's kind of hard to solve so I'm just gonna give you the solution Okay, good do that. What you do is you tell everybody in the hotel. Okay, listen up What you do is you get a list of all the prime numbers and when cool property of the prime numbers is that a prime number to a power is never gonna be equal to a different prime number to any other power, right? Right because like you can't factor it out by definition So what you do is you start with the first prime number, which is two you tell everybody in the hotel
00:16:16
Speaker
Now you take the first bus and you give them the second prime number which is three and you tell them okay first person off the bus you do three to the first second person three squared and so on okay and third bus you do that with five fourth bus you do it with seven and so on every prime number yeah and everybody will have a unique number at this point right right for sure now what you do is you tell everybody okay
00:16:41
Speaker
And of course there's gonna be a ton of empty spaces, but basically you tell everybody at the next point This is crazy and also this hotel better be free Do you know who came up with this hotel concept? No, let me look that up. Oh
00:17:03
Speaker
Oh, Hilbert, actually. I didn't know that. Hilbert's paradox of the Grand Hotel. Do you know Hilbert? Yeah, Hilbert's dope. Hilbert, he's dope. He's like a logician. He fought a little bit of the Goodell he lost, but he still had a lot of cool ideas. What timeframe? Like early 1900s.
00:17:20
Speaker
Okay. And yeah, so that's the infinity hotel.
Godel's Incompleteness Theorem
00:17:22
Speaker
Now this infinity bus is each with the infinity people. That is like Omega squared because it's Omega times Omega. Okay. And like if each one of those infinity people had a cell phone with an infinite number of contacts and each is different from the other, the number of contacts could be described by Omega cubed.
00:17:40
Speaker
Oh, wow. Okay. What's good about these omegas is that we could compare them directly because omega squared is less than omega cubed. Omega is bigger than any natural number. Omega times any natural number that's bigger than any other natural number times omega is bigger than that number. Oh, that's weird. You're taking something that's very hard to understand, to understand very simple, like very like numbers in a way, like
00:18:04
Speaker
Oh yeah, like, I mean, numbers themselves are pretty remarkably abstract. I mean, the place value system, we spent a whole episode just describing how weird that is. I mean, how cool it is, but it's weird. Right. I mean, we just have, we have 10 squiggles and we use them for everything. It's amazing. As we said before, if you count zero, one, two, three, continue indefinitely, that continuing indefinitely counting describes Omega.
00:18:27
Speaker
Yes. And so true Omega is kind of like two copies of the numbers next to each other. Okay. Like zero, one, two, three through infinity and then zero, one, two, three through infinity. Okay. I can accept that. Three Omega is three copies. Sure. Omega squared is Omega copies.
00:18:42
Speaker
Okay, okay. Well, super quick, let's talk about what omega to the omega would be. And omega to the omega would be is if you had an omega times omega times omega, omega times. Wow. And that's not even the biggest omega. There's also omega to the omega to the omega to the omega to the omega. Isn't that what you just said? No, no, I just said omega to the omega. There's a stack of infinite omegas.
00:19:05
Speaker
Omega to the Omega to the Omega, just keep going. If you have every collection from one, two, three, you have a set with one, set with two, set with three, set with Omega, set with Omega plus one, set with Omega squared, set with Omega to Omega and every number less than it. Even though Omega is always countable, that set is itself uncountable.
00:19:25
Speaker
Yeah. Yeah. Okay. That makes sense to me. I know that. I mean, it sort of does and doesn't, but I get it. I think I get it. Yeah. It's, it's uncountable in the same way that the real numbers are uncountable. Right. I understand that. And the real numbers are uncountable because like, uh, there's, there's no way to put them, uh, in order, basically. Okay. Like you're always going to, yeah. One cool thing about the trans finite ordinals and it's something that I think that we should do a quick proof sketch of, um, it's not really a proof sketch. It's more of a convince ourselves it's true kind of thing. What's a proof sketch?
00:19:53
Speaker
A proof sketch is like a lazy proof. Oh, okay. Got you. But it's not even that really, it's more of a intuitive. It's just a thought process kind of thing. Yeah, because on the next problem episode, I want to cover how this is done.
00:20:06
Speaker
It's a thought experiment. So basically what's surprising about the, about the ordinals, right? Is that if we have, if we start at any ordinal number, right? Okay. It could be, it could be 130. Whatever. It could be omega. It could be a hundred omega, whatever. Okay. If it starts there and it decreases, strictly decreases. Okay. We can count it. Yeah. It's always finite. Right. That makes sense. And I'm just wondering, how come that makes sense to you?
00:20:32
Speaker
I think because we're counting backwards. Like, so we basically started at some place and we're counting backwards. So in my mind, it's makes more sense. Yeah. Because like, let's say, let's say, let's start with, let's say we start with Omega, right? Yeah. A number less than Omega has to be a natural number. Yeah. So like it exists. Yeah. And then once we're at that number, whether it be a million or 37 or 286,543, it's a finite number of things.
00:20:57
Speaker
Okay. So let's say we started Omega plus one, right? Okay. Same difference. It goes Omega plus one to Omega to whatever. Right. And even, let's say we even start at like three times Omega, right? It's still the same because you still could backtrack it. Yeah. Cause you got to go to two times Omega plus some natural number to be less than three Omega, right? Exactly. And then that goes down to two Omega and then that goes to Omega plus some natural number or just get back to Omega. And then of course then, okay.
00:21:24
Speaker
Yeah. And basically that's the whole concept of it. And what's interesting is about, is that using this concept, we can prove a property of the natural numbers that is impossible to prove using without using infinities. Okay. And that's Goodstein's theorem.
00:21:42
Speaker
So Ruben Goodstein was an English mathematician and he he was just a logician. He was like a logician. He was cool about him is that he he studied our friend the Ackerman function.
Goodstein Sequences and Transfinite Ordinals
00:21:55
Speaker
But he also that's so funny. I think the same that actually makes sense to me. People that would delve in this would obviously be interested in all that. Those functions we were talking about.
00:22:03
Speaker
I know it doesn't actually make sense what you're saying, but it totally makes sense what you're saying. But I don't actually know why it doesn't make sense, but it just, it seems like it's not a cohesive train of thought, but it definitely makes sense.
00:22:14
Speaker
I think it's just because it's delving into these concepts of infinite and enormous. Yes, exactly. Okay. Yes. Let's talk about piano arithmetic, which is covered on the show before. Piano arithmetic is basically saying that there's a few, um, there's a few rules in piano arithmetic. Basically zero is a natural number. That's the definition definition. Number one.
00:22:36
Speaker
Every natural number is equal to itself. That's definition two. If you have two numbers, if x equals y, then y equals x. That's definition three. So it's symmetric, equality-wise. If you have x, y, and z, and x equals y, and y equals z, then x equals z. Basically, it's kind of the same stuff that Euclid did, actually, kind of. Doesn't that kind of remind you of Euclid? It just sounds like real numbers. The difference between those two is that we haven't defined any number besides zero yet.
00:23:02
Speaker
So yeah, you're right. So it more sounds more like Euclid. Yeah. Okay. That makes sense. Like basically congruence idea. Yeah. Yeah. Like, like, like subtracted from likes are like in things like where you're not actually naming e-values. Okay. Gotcha. Yeah. Or like things that coincide are equal, that kind of thing. Okay. Um, so now, uh, but this is piano. Yeah. Piano. We're with the tick P E N O. Okay. He's a, he was an Italian logician.
00:23:26
Speaker
Okay, you're going to bring that in to talk about trans-finite ordinals? Oh, well, this system that I'm describing right now, it's impossible to prove the theorem that we're going to talk about in this system. So let's say that you have some number A, right? And it's equal to a natural number B, right? Okay. It must, that means that A is a natural number. Right. That's another definition. So they're pretty basic. For every natural number, the successor to the natural number is a natural number. So we define the term successor there.
00:23:55
Speaker
Okay, sure. The successors are equal only if and only if the numbers be equal. Sure. And the successor of any natural number is not zero. Sure.
00:24:05
Speaker
because natural numbers start at zero so right those are the definite and also the last definition is a definition of induction basically saying so then the induction theorem basically says that like if i want to prove that every domino falls using induction i say i push over the first domino okay this is just another property of this okay of this piano whatever yeah sorry you couldn't remember what you said after the piano piano arithmetic is that what you're saying
00:24:30
Speaker
Oh, yeah. And super quick for the audience, induction is, uh, when you, like, if I want to prove that every domino will fall down, then I prove that the first one knocks down the second one. So you kind of have like the first link label at zero. Um, so zero is in the set K of dominoes that fall down. And then if some domino falls, then the next one is going to fall. So that's the second thing that we know. And using those two things and the axiom of induction, we could say that all dominoes fall.
00:24:59
Speaker
Sure, sure, sure. You're inducing it. Okay, I got it. Yeah. And so that has to be defined in piano arithmetic by either sets or predicates. Predicates is actually a little bit easier. So a predicate is like a test. So if the test for zero is true, and for every natural number, the test of n implies that the test of the successor of n is true, then the test is true for every natural number.
00:25:22
Speaker
Okay, yeah, I can accept that. So, Goodstein's Theorem happens to be a theorem that you cannot prove in the system, which was not discovered until the 70s. Piano arithmetic was developed in the early 1800s.
00:25:33
Speaker
And you're saying that the theorem wasn't discovered? Late 1800s. The theorem wasn't discovered or the fact that it couldn't be proved. Well, I mean, okay. So the history of this is actually kind of interesting. So in 1879, he came up with all these axioms. And then in like 1911, I believe it was Goodell released the Goodell in completeness theorem saying that basically any system of axioms will have statements that are true within the axioms, but cannot be proven with the axioms. Okay. Okay.
00:26:03
Speaker
So basically every system of axioms is either incomplete or inconsistent. Is that Godel's theorem? Yep. And inconsistent meaning would mean that you could prove anything to be true or false. It's much more useful for a system to be incomplete, because then it's at least consistent. So then basically in the 1970s, somebody came up with an actual theorem that you cannot prove using piano arithmetic.
00:26:31
Speaker
Oh, okay. Okay. And that's where this comes into play. Yep. So basically let's say, so to do a good scene sequence, let me, let's just show what a good scene sequence, how to do it is, right? Let's say we start with a number 10, right? 10 is the first number of our good scene sequence. Okay. That's a two cubed plus two, right? Yes. And now two cubed has the number three in there and we're trying to get rid of anything, but a twos and ones. Sure. Three is two plus one, right? So you have two to the two plus one.
00:27:01
Speaker
Okay.
00:27:01
Speaker
plus two. And so now let's say we for some reason replace the twos by threes. There's no reason to do this, but let's do it. Okay. And then what we then what we do is we represent it in base three hereditary representation. Hereditary representation is like the only use the base or the number one. Okay. So, so the number three to the three plus one plus three, because there comes from two to the two plus one plus two, right? We start with two to the two plus one plus two, correct?
00:27:31
Speaker
to the two plus one plus two, yes. And then we've replaced the twos by threes, we get three to the three plus one, or three to the fourth, right? Plus three, which is 84. Now, if we convert that number 84 into a hereditary base three notation, and then we keep doing that base three, base four, base five, base six, the question Goodstein asked is, do these sequences terminate? Which sequences terminate? And it seems like it wouldn't, right? It seems like it would just get bigger and bigger, right?
00:28:00
Speaker
Well, okay. So if you're taking the twos and then you replace them by threes and then you take that number and then you convert that to hereditary base three notation. So, um, and then what do you do from there after you convert it to hereditary, then you change that three to a four. Yes. And then you get in the, in each time you subtract one as well. Um, so, um, it's a very contrived example and it's famous for being contrived. It's also very just, okay. Contrived is exactly a good word for it. Nevermind. I was trying to think of the word. Yeah.
00:28:29
Speaker
So basically what you find is that these sequences always end. They do? Uh-huh. And the reason, and the way that this was proven was he replaced the twos with omegas.
00:28:40
Speaker
Oh, okay. Okay. And then he proved that each time you replace the threes with omegas, then you replace the fours with omegas. Okay. Got it. And then he proved that this sequence is strictly decreasing. Oh my goodness. And since it starts, yeah. And since it starts at a, um, ordinal number and it decreases strictly, then it's finite. That it's finite. That is so weird. That's weird. So basically he used this concept of omegas with the infinite hotels.
00:29:06
Speaker
Yeah, he used this bizarre concept to prove this bizarre theorem to prove that piano was not as good as math. No, I was kidding. No, it's not about that. It's just about proving.
00:29:17
Speaker
So basically it's a way to show that Godel's incompleteness theorem applies to piano arithmetic as any other system. It's just a confirmation of the fact or a concrete example. So constructivist mathematicians would probably like it
Reflections on Abstract Mathematics
00:29:29
Speaker
a lot. It's still really cool. I mean, it's really, really cool to think that you use something that's very hard example of understanding all the omegas and whatnot. And once you get that, then you prove that something is finite. I mean, that's kind of.
00:29:41
Speaker
And the weird thing about this good scene sequence too is that you can do it in piano arithmetic, right? You can calculate each number using piano arithmetic, but the theorem itself cannot be proven within piano arithmetic. Which is true to Godel's theorem, isn't it? Yep. Okay, well then there you go.
00:30:02
Speaker
Cardinals describe collections of things that may be infinite by labeling them. Thus we know where many is infinity, but how many is infinity? The answer, and a better way to phrase the question, on the next episode of Breaking Math, all about trans-finite Cardinals.
00:30:18
Speaker
This is Sophia, and this has been Breaking Math. With me we head on math teacher Diane Baca from APS. Thank you. Thank you for having me. It was fun. Anything you want to plug? No, thank you. Yeah, and any reflections on the episode?
00:30:37
Speaker
Um, just that it started out super frustrating. I mean, in, in a way, it's very, very like, it's a mind game. Oh yeah, definitely. You know, and I feel like it, it was kind of cool where it ended up and that something so, so, so frustratingly, you know, like, or not, I shouldn't say frustrating. That's probably a bad word, but something that was kind of so abstract ended up proving something so.
00:31:04
Speaker
So real and concrete. Yeah. I mean, as concrete as good scene sequences can be. Yeah, exactly. Okay. Thank you for being on the show.