6 Paradoxes That Will Break Your Brain

We’ve got Liam today with us who’s our excellent videographer. He does all of our videos. >> Hello. >> And Liam, do you know why you’re here today? >> I was told that we are going to be talking about some paradoxes. >> Paradoxes. That’s right. And I love paradoxes because they show the limit of human thought and they can also teach us about ways we can avoid biases and logic in our own thinking. All right. So, I

Want you to pick a first paradox for my shirt. >> Let’s go with the liars paradox. >> The liars paradox is right here. >> This sentence is false. >> Right. So this sentence is false. Now it’s almost the most basic quintessential paradox because it is if it’s true that this sentence is false well then it’s false and if it’s false that the sentence is false when it’s true. So some people want to say that it’s neither true nor false. Either

Truth or falsehood would cause a contradiction. So it must be neither. But the problem is even if you say it’s neither true nor false you can actually just redefine it slightly and still produce a paradox. >> What do you mean by if you redefine it slightly? Well, suppose instead of saying this sentence is false, we said this sentence is not true. Well, if we said this sentence is not true, well, if it’s neither true nor false, then that

Is true about it that it’s not true and then you can still get a paradox. Now, an issue that people have with the liars paradox is they say, well, maybe the whole thing is it’s recursive. It’s it’s referring to itself. >> Maybe we should just disallow that. Like when things refer to themselves, they can get wonky. They can get paralaxical. So maybe the resolution just say don’t allow things to refer to themselves. But that is where my boy Diablo comes into

Play and he produces the Diablo’s paradox which is right here. Okay, so this says all the sentences below this are false and then the next line is all the sentences below this are false and it just seems to be continuing on down and down. >> So let’s think about that for a second. Suppose that the first sentence is true. What are the implications of that? That means that the second sentence has to be false. And so what does that imply?

So then that would mean that not all of the sentences are false. >> Exactly. >> If the second statement is false, then at least some of the statements below it are true. >> Yeah. >> But that’s a contradiction because we know if the first sentence is true, that contradicts it. So Yablo basically produced this paradox saying this is not about recursion fundamentally because none of these statements refer to themselves. They all refer only to the

Statements beneath them. This kind of paradox is not unique to recursion. Okay. So this brings us to another paradox or from the barber. >> So this is the barber. >> Say hello to the barber. >> He’s not looking too good. Now the barber, he has this really unique property which is that he shaves everyone that doesn’t shave themselves. Everyone who doesn’t shave themselves and only people who don’t shave themselves are shaved by the barber.

Okay. >> Now the barber has a question for you. Should he shave himself? Obviously, everybody in town that doesn’t shave themselves is going to get shaved by him. They have to get shaved by him, >> right? So, suppose that he does shave himself, >> then he’s not within like that set of people that can be shaved by him. >> Exactly. If he shaves himself, then he’s someone who shaves himself, so he can’t be someone he shaves. So, you have a

Contradiction. But if he doesn’t shave himself, then he’s someone he’s supposed to shave, and that’s also a contradiction. So, this might seem a bit like the liar paradox. >> Mhm. Uh but what’s interesting is instead of a sentence referring to itself, it’s setting up this construct of the barber that seems to have no way of acting in a non-contradictory manner. But that’s what brings us to the real challenge here, which is Russell’s paradox, which is the equivalent of the

Barber paradox, but in math. And the problem is you can’t just say, well, the barber doesn’t exist. Uh because what we’re talking about here is a mathematical set. Russell’s paradox says consider the set that contains all sets that don’t contain themselves. And so you can ask the question, well does it contain itself? And it unfortunately puts us in exactly the same situation as the barber because if it contains itself, well then it contains a set that

Contains itself which it’s not supposed to do. >> But if it doesn’t contain itself, then it’s missing a set it’s supposed to contain, right? So this idea, this Russell set of the set of all sets that don’t contain themselves, we can’t answer the question of whether it contains itself, which seems like a paradox. Now the problem is this is a mathematical set that we’re defining. And so they you can’t just say, well, it doesn’t exist. You either math implies

That it exists because it’s part of math or it doesn’t follow from the axioms of math in which case it can’t be constructed. And actually to resolve this it produced some interesting work where mathematicians had to be very careful and to say well how are we actually defining things we have to find them in a really careful way to avoid potential paradoxes. This is actually a good example of how thinking about paradoxes can actually force us to

Improve our thinking overall and to get really precise. Talking about the idea of precision actually brings us to our next paradox which is the heap. So Liam, would you say that this is a heap of candy? I would say that’s a heap of candy. >> Now, despite them being ridiculously small candies, I would also agree with you that this is a heap. Let me give you one of these. >> All right. Thank you. >> Would you agree that this bag still has

A heap? >> I would say it’s still a heap of candy. >> Okay, cool. So, you’re saying basically if you’ve got a heap of candy, you remove one, you still have a heap of candy. >> Yes. >> Okay, cool. Now, show me the candy I gave you. Now, would you say that this candy on its own is a heap? >> No, this is not a heap of candy. >> Okay, but here’s the problem. You just contradicted yourself because you said

That if you had a heap of candy, you removed one, it’s still a heap. >> Yeah. >> But if we kept doing that, we’d eventually get down to one candy, which is saying it’s not a heap. >> Yes. Yeah. I mean, there’s probably a number or a certain point where I would stop thinking that something is a heap. But I probably wouldn’t say that there’s a point where I would say something’s a heap and I could just take one away from

It and that would like stop it from being a heap. >> Yeah. So I think in practice, let’s say I was doing this for every candy. I removed one, said, “Is it heap?” Removed one, is it heap? I think eventually we get down to a point where you’d start to be more uncertain. You’d be like, “Well, I don’t know.” And then eventually you’d say, “No, it’s not a heap anymore, right? It’s some number.” Yeah. >> The thing about that is that you

Probably don’t know what that number is. So we could say that this might be a paradox of ambiguity. On the one hand, there’s ambiguity about what people in general would mean by a heap, right? If you were to ask different people, you could get different answers. But I would argue there’s an even deeper ambiguity which is that individuals don’t even know what they mean by a heap. It’s not like, you know, whenever we use a word,

We’re like secretly thinking about its definition. Instead, we just have a cluster of associations and concepts that all work together, right? For example, if you think about a hippopotamus, you might imagine a particular hippopotamus you saw maybe at the zoo. You might be incorporating photographs of hippopotami you’ve seen. Maybe you you’ve heard things about hippopotami and maybe you’re incorporating those as well. So the way the human mind works is we sort of have this concept which is a cluster of

Different stuff. Images, sounds, associations, facts all working together. We don’t have a precise definition. When we say is it heap or is it not a heap, we don’t actually know what we mean by that. We just have this association with the idea of heap. Now in a certain way, I would say this is kind of a paradox. It’s I don’t know if it’s so fundamentally paradoxical as it is dealing with ambiguity, but what I think is cool

About it is it actually raises questions that come up in everyday life. So, are there any paradoxes on here that you recognize? >> Um, I recognize Zeno’s paradox with the turtle. >> Yes. So, let’s do Zeno’s paradox. And let’s start with a race. So, this is Zeno’s race. >> All right. >> And you’re going to be the hair >> and I will be the tortoise. And here’s the idea. We have to start at the start and we have to move our guys to the

Finish. But there’s a catch. >> And the catch is because it’s Zeno’s race, we have to make sure that we touch the halfway point between the start and the finish. But we also have to touch the point that’s halfway between this halfway point and the finish. And then we have to touch the point that’s halfway between this halfway to the halfway point and the finish >> and so on and so forth. >> So each step is just halfway towards

However far the finish line is. >> Exactly. Exactly. So you got the rules. Okay, let’s race. Ready? On three. 1 2 3 go. I touched every single point. I touched this halfway point, halfway to the halfway, halfway, halfway, and so on forever. So Zeno used this idea to argue that there is no motion. Because he said, well, look, in order to move between two points, like the start and the finish, you first have to go halfway, and then once you’re at the

Halfway point, you have to go halfway from there to the finish. And then once you’re at that point, you have to go halfway from there to the finish, etc. There’s an infinite number of those points. There’s no way you can take an infinite number of actions. So there’s no way you can move from the start to the finish. Therefore, motion is impossible. >> Might be a little controversial. >> Yeah. Well, it seems like we move all

The time, right? So, this is a funny kind of paradox where we know that the conclusion is false, but then the question is what’s wrong with the logic, right? Can we actually do an infinite number of actions? One way you could resolve this is saying, well, yes, in theory, you could divide it infinitely like that, but maybe the physical reality doesn’t work that way. Maybe physical reality is not infinitely divisible and therefore there’s not an infinite number of actions. Another way

To resolve this is to say actually you can take an infinite number of actions. Well, is that actually true? Here’s the thing. Let’s figure out how long each action takes. Let’s suppose the race were a little longer and it would take you 16 seconds to go from the start to the finish. >> It’s 8 seconds to do the first action plus half of that which is four which is 12 seconds plus another half of that which is 2 which is 14 seconds and so

On. If you actually add up that infinite series which has an infinite number of terms to it, you find it actually does sum to 16. So a resolution of the paradox is yes, you have to take an infinite number of actions, but the actions are defined in such a way that each subsequent one takes half the time and and if you add it all up, it’s still a finite amount of time. So one way to think about it is that how many actions

There are is something that we invent. Like we define what an action is. Yeah. So for any given task, we could define it in such a way that there’s an infinite number of actions, but we could also just define it differently. So there’s a finite number. Yeah. So there’s not truly in some deep sense an infinite number of actions. It’s just how you want to divvy it up, right? And you we happen to divvy it up in a funny

Way so that the number of actions was infinite. Now infinities are actually at the heart of many paradoxes. So let’s jump into another paradox about infinities. So this is Hilbert’s hotel. So, Hilbert owns this hotel, and it’s just like a normal hotel, but it’s infinite. There are an infinite number of rooms. Every single room is full tonight. Uh, but there’s a problem, which is that new visitors just arrived, and Hilbert really wants to fit them in,

but he doesn’t want to ask any guests to share a room. He doesn’t want to kick anyone out of the hotel. He can’t build a new room tonight. >> So, the question is, how does he fit in this new guest that just arrived without kicking anyone else out? The answer is that you go to, you know, the first room and you ask the guest if they can go to the second room and tell that person to

Tell the next guest to move into the next room. So now you have the first room and everybody else is moving into the room next to them. >> Exactly. That’s right. So if the people in room one moved to two and two move to three and three move to four, etc. Once you kind of propagate that all out, you find that wow, there’s a new room that’s open at one. But what’s so strange about it is that every single room was full

Before and yet somehow just by shifting you made a new room and now you can fit the guest in. I would argue that the paradox comes about because our intuition about infinity is not very good. You know we there’s a lot of things that we believe are true about the the regular world or about finite things >> that when you push yourself to an infinity they stop being true. So for example if you have a finite number of

Things there’s no way you could make an extra slot by just moving things around. >> Yeah. Right? Well, if you have 10 slots and they’re all full, shifting them is never going to make a new slot. Infinities are weird. Infinities, that’s not true. So, our intuition about these finite things doesn’t carry over. And so, it feels like a paradox. So, clearly, infinities just don’t work like the things we’re used to. And this actually ends up connecting to sort of

The study of infinities themselves. >> There’s this idea you may have heard of that there can be different sizes of infinities. So, the integers, there’s an infinite number of them. >> Yeah. But there’s some real sense in which the real numbers which include numbers like you know pi and the square root of two the size of the real numbers is bigger than the size of the integers. This idea of Hbert’s hotel actually connects to sort of very important ideas

In studying infinities. So Liam we’ve done half of the paradoxes on the shirt. If people enjoy this video we’ll do the second half. But for now which of these blew your mind the most? The Hilbert’s Hotel one really messes with me because it’s the the idea of infinity just like I don’t know it’s like something that’s actually like you know useful and like relevant to a lot of disciplines but it just like it feels it feels really wrong

To think about. >> Yeah. It’s so funny because in math we use infinities all the time. We don’t know if infinities exist in the real world. Like we don’t know maybe the universe is infinite in space. >> It’s actually unknown. A lot of people think it’s known, but it’s not known whether it is or not. In in actual real life, we don’t know if infinities exist. A lot of people think they don’t. But in math, we actually use them all the time,

And they’re incredibly useful, and they actually have these interesting, precise properties, but those properties are often not what you think because infinities are totally different than the things we encounter in the normal world. If you found this interesting, we’d love it if you’d subscribe to our channel. And if you enjoyed this video, let us know in the comments because if you like it, we’ll go do the rest of the paradoxes on the shirt.