Infinity, Paradoxes, Gödel Incompleteness & the Mathematical Multiverse | Lex Fridman Podcast #488

- The following is a conversation with Joel David Hamkins, a

mathematician and philosopher specializing in set theory,

the foundation of mathematics and the nature of

infinity. He is the number one highest

rated user on MathOverflow, which I think is a legendary

accomplishment. MathOverflow, by the way, is like StackOverflow

but for research mathematicians. He is

also the author of several books, including Proof

in the Art of Mathematics and Lectures on the

Philosophy of Mathematics. And he has a great blog,

infinitelymore.xyz. This is a super technical and

super fun conversation about the foundation of modern mathematics

and some mind-bending ideas about infinity,

nature of reality, truth, and the mathematical

paradoxes that challenged some of the greatest minds of the 20th century.

I have been hiding from the world a bit, reading, thinking, writing, soul-searching,

as we all do every once in a while. But mostly, just deeply focused

on work and preparing mentally for some challenging travel I

plan to take on in the new year. Through all of it, a recurring thought comes to me,

how damn lucky I am to be alive and to get to

experience so much love from folks across the

world. I want to take this moment to say thank you

from the bottom of my heart for everything, for your

support, for the many amazing conversations I've had

with people across the world. I got a little bit of hate

and a whole lot of love, and I wouldn't have it

any other way. I'm grateful for all of it.

This is the Lex Fridman Podcast. To support it, please check out our

sponsors in the description, where you can also

find ways to contact me, ask questions, give feedback, and so on. And now, dear

friends, here's Joel David Hamkins.

Some infinities are bigger than others. This idea from

Cantor at the end of the 19th century, I think it's fair to

say, broke mathematics before rebuilding it. And

I also read that this was a devastating and transformative discovery for

several reasons. So one, it created a theological

crisis. Because infinity is associated with God, how could there be multiple

infinities? And also, Cantor was deeply religious

himself. Second, there's a kind of mathematical civil

war. The leading German mathematician Kronecker called Cantor a corrupter of youth

and tried to block his career. Third, many

fascinating paradoxes emerged from this, like Russell's

paradox, about the set of all sets that don't contain themselves,

and those threatened to make all of mathematics

inconsistent. And finally, on the psychological side and the personal

side, Cantor's own breakdown. He literally went mad,

spending his final years in and out of sanatoriums, obsessed with

proving the continuum hypothesis. So, laying that all out on the

table, can you explain the idea of infinity, that some infinities are

larger than others, and why was this so transformative to mathematics?

- Well, that's a really great question. I would want to start

talking about infinity and telling the story much

earlier than Cantor, actually, because, I mean, you can go all the way

back to Ancient Greek times when Aristotle

emphasized the potential aspect of infinity as opposed

to the impossibility, according to him, of achieving an actual

infinity. And Archimedes' method of exhaustion where

he is trying to understand the area of a region

by carving it into more and more triangles, say, and

sort of exhausting the area and thereby understanding the total area in terms

of the sum of the areas of the pieces that he put into it.

And it proceeded on this kind of potential understanding of infinity for hundreds of years,

thousands of years. Almost all mathematicians were

almost all mathematicians were potentialists only and

thought that it was incoherent to speak of an actual

infinity at all. Galileo is an extremely prominent exception to this, though he

argued against this sort of potentialist orthodoxy

in The Dialogue of Two New Sciences. Really lovely

account there that he gave. And that the... In many ways, Galileo was anticipating

Cantor's developments, except he couldn't quite push it all the way

through and ended up throwing up his hands in confusion

in a sense. I mean, the Galileo paradox is the idea or the

observation that if you think about the natural numbers,

I would start with zero but I think maybe he would start with one. The numbers one,

two, three, four, and so on, and you think about which of those

numbers are perfect squares. So zero squared is

zero and one squared is one and two squared is four, three

squared is nine, 16, 25, and so on. And Galileo

observed that, that the perfect squares can be

put into a one-to-one correspondence with all of the

numbers. I mean, we just did it. I associated every number

with its square. And so it seems

like on the basis of this one-to-one correspondence that

there should be exactly the same number of

squares, perfect squares as there are numbers, and

yet there's all the gaps in between the perfect squares,

right? And, and this suggests that there should be fewer perfect squares, more

numbers than squares because the numbers include all the squares plus a lot more in

between them, right? And Galileo was quite troubled by this observation because

he took it to cause a kind of incoherence in the

comparison of infinite quantities, right? And another example

is, if you take two line segments of different lengths,

and you can imagine drawing a kind of foliation,

a fan of lines that connect them. So the endpoints are

matched from the shorter to the longer segment, and the midpoints are matched and

so on. So spreading out the lines as you go. And so every

point on the shorter line would be associated with

a, a unique distinct point on the longer line in a

one-to-one way. And so it seems like the two

line segments have the same number of points on them because of that, even

though the longer one is longer. And so it

makes, again, a kind of confusion over our ideas about infinity. And

also with two circles, if you just place them

concentrically and draw the rays from the center,

then every point on the smaller circle is associated

with a corresponding point on the larger circle, you know, in a

one-to-one way. And, and again, that seems to show that the

smaller circle has the same number of points on it as the larger one, precisely

precisely because they can be put into this one-to-one correspondence.

Of course, the contemporary attitude about this situation is that those two

infinities are exactly the same, and that Galileo was right in those

observations about the equinumerosity. We would talk about it now

by appealing to what I call the Cantor-Hume principle, or some people

just call it Hume's principle, which is the idea that if you have two

collections, whether they're finite or infinite, then we want to

say that those two collections have the same size. They're

equinumerous if and only if there's a one-to-one

correspondence between those collections. Galileo was

observing that line segments of different lengths are equinumerous,

and the perfect squares are equinumerous with all of the

natural numbers, and any two circles are equinumerous, and so

on. The tension between the Cantor-Hume

principle and what could be called Euclid's principle, which is that the

whole is always greater than the part, is a principle that Euclid

appealed to in the Elements many times when he's calculating

area and so on. It's a basic idea

that if something is just a part of another thing, then the

whole is greater than the part. So what Galileo

was troubled by was this tension between

what we call the Cantor-Hume principle and Euclid's

principle. It wasn't fully resolved, I

think, until Cantor. He's the one who really explained

so clearly about these different sizes of infinity and so on

in a way that was so compelling. He

exhibited two different infinite sets and proved that

they're not equinumerous; they can't be put into one-to-one

correspondence. It's traditional to talk about the

uncountability of the real numbers. Cantor's big result was that the

set of all real numbers is an uncountable set. Maybe if we're

going to talk about countable sets, then I would suggest that we talk

about Hilbert's Hotel, which really makes that idea perfectly clear.

- Yeah, let's talk about Hilbert's Hotel.

- Hilbert's Hotel is a hotel with infinitely many rooms. Each

room is a full floor suite. So there's floor zero... I always

start with zero because for me, the natural numbers start with zero, although that's

maybe a point of contention for some mathematicians. The other

mathematicians are wrong.

- Like I mentioned, I'm a programmer, so starting at zero is a wonderful place to start.

- Exactly. So there's floor zero, floor one, floor two, or room zero,

one, two, three, and so on, just like the natural numbers. So Hilbert's Hotel has a

room for every natural number,

and it's completely full. There's a person occupying room N for

every N. But meanwhile, a new guest comes up to the

desk and wants a room. "Can I have a room, please?" The manager says,

"Hang on a second, just give me a moment."

You see, when the other guests had checked in, they had to sign an

agreement with the hotel that maybe

there would be some changing of the rooms during this stay.

So the manager sent a message up to all the current

occupants and told every person, "Hey, can you

move up one room, please?" So the person in room five would move to

room six, and the person in room six would move to room seven and so on. And everyone

moved at the same time. And of course, we never want to be placing two different

guests in the same room, and we want everyone to have their own private room

and... But when you move everyone up one room, then the bottom

room, room zero, becomes available, of course. And so he can put the new guest in

that room. So even when you have infinitely many

things, then the new guest can be accommodated. And that's a way of

showing how the particular infinity of the occupants of

Hilbert's Hotel, it violates Euclid's principle. I

mean, it exactly illustrates this idea because

adding one more element to a set didn't make it larger, because we

can still have a one-to-one correspondence between the total new

guests and the old guests by the room number, right?

- So, to just say one more time, the hotel is full.

- The hotel is full.

- And then you could still squeeze in one more, and that breaks

the traditional notion of mathematics and breaks

people's brains when they try to think about infinity, I suppose. This is

a property of infinity.

- It's a property of infinity that sometimes when you add an

element to a set, it doesn't get larger. That's what this

example shows. But one can go on with Hilbert's

Hotel, for example. I mean, maybe the next day, you know,

20 people show up all at once. We can easily do the same trick again, just move

everybody up 20 rooms. And then we would have 20

empty rooms at the bottom, and those new 20 guests could go

in. But on the following weekend, a giant bus pulled up, Hilbert's

bus. And Hilbert's bus has, of course,

infinitely many seats. There's Seat Zero, Seat One, Seat Two, Seat Three, and so

on. And so one wants to... You know, all the people on the bus want

to check into the hotel, but the hotel is completely full. So what is the manager going to

do? And when I talk about Hilbert's Hotel,

when I teach Hilbert's Hotel in class, I always demand that the

students provide, you know, the explanation of- of how to do it.

So maybe I'll ask you.

Can you tell me, yeah, what is your idea about how to fit them all in the hotel, everyone

on the bus, and also the current occupants?

- You separate the hotel into even and odd rooms, and you squeeze in the new

Hilbert bus people into the odd rooms and the previous occupants