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.

>> 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.