## Math Without Numbers - Infinity

Go back to the Math Jam ArchiveIs anything bigger than infinity? What about infinity plus one? Guest speaker Milo Beckman, NYC Math Team alum and author of the newly-released book Math Without Numbers, will lead a discussion on one of the most peculiar and surprising topics in modern math: infinite cardinalities. We'll talk about what it really means for a quantity to be "bigger" than another, and use that knowledge to test out some different things that might be bigger than infinity. (And we won't be using any numbers.)

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#### Facilitator: AoPS Staff

Hello and welcome to the

**Math Without Numbers Jam**!

hi

hi

hi

hi

Salve!

HI!!

hi

yay!

Now let me introduce our guest for the evening.

Milo Beckman (

**miyomiyo**) is an NYC Math Team alum and author of the newly-released book Math Without Numbers.

Thanks for being here with us this evening, Milo!

Hey everyone! Welcome to a Math Jam... without numbers!

That's right — a lot of the math people do in college and beyond doesn't involve any numbers at all. I remember realizing once, in a college math class, that the blackboard was filled with equations and diagrams and there wasn't a single number anywhere.

It inspired me to write a whole book about math without using any numbers. It just came out this week and I'm really excited for everyone to finally read it! If you're interested you can order it here: https://www.penguinrandomhouse.com/books/609844/math-without-numbers-by-milo-beckman/

Today I want to give you a little preview of one of my favorite topics in abstract math, without using any numbers.

The topic is: infinity!

(Technically the topic is "infinite cardinalities and Cantor's diagonalization argument," but let's just say "infinity" for now.)

We're going to be focused on one big question: Is anything bigger than infinity?

Before we even get started, I want to hear your guesses. Just from your intuition, do you think there's something bigger than infinity?

Nope

nope

yes

NOPE!

Yes

Hmm...

No

No

NOPE

yes

Ok, interesting. We'll figure out the answer by the end of this Math Jam, but you're going to have to help me get there.

The first step is defining our terms. When you do math without numbers, you have to be really precise about what all the words you're using actually mean. Otherwise the question is too vague and you can't get a solid answer.

For our question "Is anything bigger than infinity?" - what words do you think we need to define?

Infinity

"Infinity"

infinity

infinity

infinity

"Bigger"

infinity

bigger and infinity

infinity

Infinity and bigger

Yes, we're definitely going to want to define "infinity." But first (this may seem ridiculous) but I want to focus on a different word: "bigger."

What does it mean when we say one thing is "bigger" than another?

I know, it's really obvious, at least when we're dealing with numbers. But we're working with infinity today, so we need to be extra careful.

Let's come up with a rule—a totally solid, foolproof rule—that tells us when one quantity is "bigger" than another.

Well, how do we define "bigger" in a normal, finite context? What does it mean to say that the amount of objects in the right pile on is "bigger" than the amount on the left?

You have to imagine you're talking to some alien from another planet, someone who's never heard of the concept of "bigger" or "more" or "greater" or anything like that. How would you explain this concept to them?

It's such a basic, obvious concept that it's actually pretty hard to describe from scratch.

Let's use a common math trick: try asking the exact opposite question. What does it mean to say that these two piles are the same size?

they have an equal amount of objects

They have the same amount of stuff

they have the same number of "things"

They are identical in amounts of items.

Again, we can't lean on the word "equal" because that's exactly what we're trying to define. The alien you're talking to has never heard of the concept of "equal" either.

Any other ideas? Let's think about it for a minute.

We can pair them off equally

you can pair up objects exactly with nothing left over on either side

each item is able to be paired up

That's a great idea! We can tell the piles are equal because we can match them up perfectly without any leftovers.

That's a good general rule to use for deciding whether two quantities are "equal" or "the same size." Can we match them up without leftovers?

Look, I'll make it into a sticky:

Two piles are "equal" if you can match up their objects without any leftovers.

Ok, now let's flip it back around. Can anyone use this to come up with a rule for when one pile is "bigger" than the other?

so if they aren't equal then after pairing the objects up one pile will have some leftovers

if the pile has leftovers then it is bigger

After matching the objects up, the group with leftovers is bigger than the other one.

When there is a remainder after pairing.

There we go! That's a great definition for the word "bigger." Let's sticky that too.

If you can't match up two piles perfectly, the side with leftovers is the "bigger" pile.

So now that you know what "bigger" means, does anyone want to change their guess? Do you think there's something bigger than infinity? Yes or no?

No

No

yes

no

no

Nope

yes

Yes.

no

I'm still not going to tell you the answer just yet. We have to figure it out.

When we talk about infinity today, we're talking about an infinite pile of objects. Not a billion or a quintillion or a Graham's number of objects, but really INFINITE.

The technical definition is: It's the size of the set of all natural numbers.

Think of it as a bottomless bag of objects. No matter how many you take out, there's still an infinite amount left in the bag.

How could anything be bigger than that?

infinity + 1?

+1

+ 1 more?

Yeah, what about infinity plus one?

It doesn't seem like one extra object should make any difference, but let's use our "bigger" rule just to make sure.

First I'm going to arrange our objects in a line so they're easier to keep track of.

If we try to match up the objects in the obvious way, it certainly looks like infinity plus one is bigger...

But is there a way to match them up without leftovers?

You can shift infinity to the right to match up

Yes! match up the extra first

if you scooch one back?

Yep, there is. Check it out:

Does that seem like cheating? Think about it for a minute. Every object has a match! Since both bags go on forever, you're never going to find an object without a partner, no matter how far down the line you go.

So the two piles are the same size! Infinity plus one equals infinity.

That's pretty bizarre, right?

what the

Cool

Here's another way of looking at the same problem. This is a famous example, called Hilbert's Hotel.

Imagine you’re the receptionist at a very special hotel, which has infinitely many rooms. There’s one long hallway, with a line of doors, and the doors go on and on forever. There’s no “room number infinity” or “last room” because there’s no end to the hallway. There’s a first room, and then for every room, there’s a next room over.

Tonight is an especially busy night: Every room in the hotel is full. (Yes, this world has infinity people, too.)

Then someone walks into the hotel lobby from the outside world and says, “Could I please have a room?”

What do you do? Do you tell them the hotel is full and turn them away?

Just move everyone to the right one room

shift everyone up by one

everyone moves to the next room

move everyone down to room number + 1

you would ask them all to move to n + 1 room

do you move everybody over 1

Right! You get on the PA system and make an announcement: “Apologies for the inconvenience. All guests please move down one room. That’s right: Pack up your things, go out into the hallway, and relocate to one room further away from the lobby. Thank you and have a great night.”

Once everyone does what you say, you’ve cleared out a room for the new guest.

Yeah, of course this could never happen in real life, because you can't have an infinite hotel or infinitely many people. But in abstract math-world we have to deal with infinite quantities pretty often, actually.

Ok, so we're all agreed: infinity plus one is still infinity.

What about, I don't know, infinity plus five?

it's still infiinty

infinity

same thing

equals infinity

still just infinity

yep! still equal

still infinity

Yeah— it doesn't make a difference. We could use the exact same trick: just have everyone move down five doors instead.

Even infinity plus a billion is still just infinity. Any finite number added to infinity, you can match it up to infinity without leftovers.

What else could we try? What might be bigger than infinity?

What is infinity+infinity then?

Infinity plus infinity?

infinity+infinity?

what about infinity + infinity

infinty + infinity???

infinity+infinity?

but what about infinity plus infinity

what about infinity plus infinity?

infinity + infinity

There's an idea: What about infinity plus infinity?

Can two infinity bags be matched up with one?

Or, in the hotel example, can we find rooms for an infinite line of new guests?

We can't use the same "shifting over" trick. How do you shift over by infinity? Where would the person in Room One even go?

So is it impossible? What do you think?

we could tell them to go to room 2n

1 goes to 2, 2 goes to 4, 3 goes to 6

they move to their room number times 2

room one goes to room two, room 2 goes to room 4, etc etc

put people in their room number * 2

Exactly! The key is to space them out.

You get on the PA system and say: “Will the guest in the first room please move to the second room, will the guest in the second room please move to the FOURTH room, and in general will the guest in room $n$ please relocate to room $2n$?"

(Darn, I used a number. I promise it won't happen again.)

Everyone still has a room and, miraculously, you've opened up infinity new empty rooms for the new guests. Every odd-numbered room is empty.

Here's the same argument with bags of objects:

So there you have it: Infinity plus infinity equals infinity.

Are we ready to give up? Or does anyone think there's something bigger?

what about infinity times infinity

NOW FOR INFINITY TIMES INFINITY

what about infinity * infinity though?

Can we try $\infty \times \infty$?

Infinity times infinity!

what about infinity*infinity?

Inifnity times infinity.

Ok, what about infinity TIMES infinity?

So... something that looks like this:

An infinite grid, with infinitely many infinitely long rows of objects. Is THAT bigger than infinity? Let's hear some guesses: bigger, or still equal?

equal

equal

equal

equal

hmm

equal

still equal

Equal!!!

bigger

equal?

equal?

=

Equal

bigger

same?

still equal

Bigger?

equal

equal

Can anyone prove their answer is right? Guessing is one thing, but I want to see some proof.

If you think infinity times infinity equals infinity, I need someone to explain to me how you're going to rearrange an infinite grid of objects into an infinite line, without leftovers.

And if you think it's bigger, well, I'm going to need to hear a really convincing argument for why you CAN'T match them up, no matter how hard you try.

Any ideas? I'll give you a minute to think about it.

I know, you can't really draw pictures here. But if you think you found a way to put all these objects in order, could you try to describe it for us in words?

The answer is EQUAL. Infinity times infinity still equals infinity.

Want to see a proof?

yes

yes!

ya

yes please

yes

YES!

This is called the "snake proof" and I think it's pretty neat.

If you "snake" your way diagonally back and forth on the grid like this, you're going to hit every single object. So we've found a way to arrange the infinity times infinity objects in a row, in an order, with no objects left out.

And if we can arrange all the objects in order, well, we can match them up one-to-one with the original infinity bag. Infinity times infinity equals infinity. QED.

("QED" is what you say at the end of a proof. It's the math equivalent of a mic drop: Boom, we proved it! It stands for Quod Erat Demonstrandum, which is Latin for "what was to be demonstrated.")

So there you go, QED. Does everyone find this proof convincing?

Ohhh..

Cool times infinity!

Oh thats pretty neat!

yeah

yes

yup

Yes!

yes

Yeah

Great, so it's settled. Infinity times infinity equals infinity.

Are we ready to give up? What do you all think: Is there something bigger than infinity?

nope

no

no

no!

No there is nothing bigger than infinity

maybe

Nothing is bigger.

maybe..

$Nope!$

Yes!!!

I'm pretty sure there is something bigger

nope, there is nothing bigger than infinity

I'm ready to tell you the answer now. We're still going to have to prove it, but I can tell you the answer. Ready?

YES.

ready!

YES!

yes

TELL US!

There IS something bigger than infinity!

It's called "the continuum."

We usually write it with a fancy lowercase c, like this: $\mathfrak{c}$

You can type it like this: \mathfrak{c}.

So what I'm claiming now, and what we're about to try to prove, is this: $\mathfrak{c} > \infty$.

But first, before we can prove anything, what IS the continuum? What is this thing $\mathfrak{c}$ that I'm claiming is bigger than infinity?

There are a lot of ways to define $\mathfrak{c}$, but here's my favorite: it's the number of points in a line.

It doesn't matter whether it's an infinite line or a finite line segment. It's the "density" that matters, so to speak. We're not dealing with separate, distinct objects anymore, like the stones in a bottomless bag. We're talking about a continuous stretch of points that blend into each other.

Let's look at this on the number line.

Which of these pictures represents $\infty$ and which represents $\mathfrak{c}$?

c is on top

the top one is c

bottom is infinity

c represents the first one

top is c and bottom is infinity

Right, the bottom one is $\infty$. There are infinitely many points (assuming the line extends forever) but they're all separate. The top is $\mathfrak{c}$ because every single point on the line is included!

So, let's check if this is making sense. How many positive integers are there? $\infty$ or $\mathfrak{c}$?

infinity!

infinty

infinity

infinity

infinity

infinity

infinity

infinity

Yep, $\infty$. And how many positive integers are there total — positive, negative, and zero? $\infty$ or $\mathfrak{c}$?

still infinity

Infinity.

infinity

infinity

still infinity

infinity

infinity?

infinity still

Still $\infty$. It's a bit weird that there are just as many positive integers as integers total, but that's what you get when you mess with infinity. Remember, $\infty + \infty = \infty$.

Ok, how many REAL numbers are there? Including fractions, square roots, transcendental numbers like $\pi$ and $e$, and everything in between?

c

$\mathfrak{c}$

now it's c

$\mathfrak{c}$

$\mathfrak{c}$

$\mathfrak{c}$

c

$\mathfrak{c}$

$\mathfrak{c}$

the fancy c

Yep, that's $\mathfrak{c}$. When every number on the number line is included, not just the spaced-out integers, you get a continuous infinity.

Ok, I think we're ready for the big proof. It's time to prove that $\mathfrak{c} > \infty$. We're going to prove that there really is something bigger than infinity.

Now how on Earth are we going to prove something like that? Let's check back on our definitions in those stickies up top.

What do we need to do in order to prove that the continuum is "bigger" than infinity?

it has leftovers when compared to infinity

That there are leftovers after pairing.

See if there's leftovers

theres leftovers after pairing

making sure there is left over on the c side

There need to be leftovers on the $\mathfrak{c}$ side, yes. But we need to show that there's NO POSSIBLE WAY to match them up without leftovers.

We need to prove that it's

*impossible*to match up the points on a line with the objects in a bottomless bag.

That's hard to do. If you want to prove something is

*possible*, that's simple enough: just do it! But if you want to prove something's

*impossible*, that's a lot trickier.

We can't just try a couple times to match them up and then throw up our hands and say, "See? Impossible!" That's not very convincing. What if your very clever rival comes along and finds a new tricky way to match them up that you didn't think of?

No, in order to prove it's impossible, we need to show that

*every conceivable matching*will fail. We need to show that the points on a continuous line segment can't be arranged into a list, not even an infinite list, no matter how hard you try.

Yes!!

yes

yea

yes!

YES

yep

Yes!

Alright, great. First things first: let's name the points. We need to have some way to refer to the points on the line, so we know what we're talking about. (And let's stick with a finite line segment, for now!)

We're going to assign every point on the line segment an "$LR$-address" in a neat, systematic way.

The first letter in the $LR$-address tells you if the point is on the left half ($L$) or right half ($R$) of the line segment.

The second letter tells you if the point is on the left half ($L$) or right half ($R$) of

*that half*.

The third letter tells you if the point is on the left half ($L$) or right half ($R$) of

*that quarter*, and so on and so on, zooming in closer and closer on the point.

But, what if its in the middle

What if it's in the middle?

Good question. If a point is exactly in the middle of the segment you're looking at, let's write an $M$ and stop the address there.

Here's an example of an arbitrary point on the line segment, and how we would start to give it an $LR$-address.

Can someone tell me what $LR$-address we would give to the point all the way on the left of the segment?

LL...

$LLLLL\dots$

LL... to infinity

LL...

LLL...LLL...

LL... repeating forever

Yep: that point is named $LLLLLLLL\ldots$ (continuing forever).

What about the point that's three-quarters of the way to the right?

RM

RM

$RM$

RM

RM

RM

RM

That one's just named $RM$. You go to the right half, and then your point is exactly in the middle.

And what about the point that's one-third of the way across? Think about it.

(This one's pretty tricky.)

LRLRLRLRLR...

LRLRLR....

LRLRLR...

LRLRLRLR...

$LRLRLRLRLRLRLRL\dots$

LRLRLR...

LRLRLRLRLRLRLRLRLRLRLRLRLRLR...

LRLRLRLRLRLR...repeating forever

LRLRLRLRL....

That one would be named $LRLRLRLR\ldots$ (repeating forever).

So every point on the line segment has a unique $LR$-address, and every $LR$-address picks out a unique point on the line segment.

What we need to do now is prove that there are too many $LR$-addresses to put into a list, even an infinite list. We want to show that

*any possible*list of $LR$-addresses is incomplete.

So let's play a game. Imagine your rival walks into the room and hands you an infinite list of $LR$-addresses, and they say, "I did it! I put all the $LR$-addresses into an infinite list." We need to prove them wrong! We need to show there's an address they're missing.

Let's be generous to your rival and say they don't even need to include any of the addresses that end in $M$. They only need to cover the "pure" $LR$-addresses, the ones that go on forever and only use $L$ and $R$. We're going to show that even

*that's*impossible.

How can we prove their list is incomplete? Any ideas?

What's the

*least*we could do to prove that this proposed list is incomplete?

make a new address

find something that is missing

find a point not on the list

get an LR addres that can't be on the list

Right, we just need to find one missing $LR$-address. That's enough to show they were wrong.

But here's the key: We need a

*systematic*way of taking any list they could possibly hand us, and finding a missing $LR$-address. We need to show that for any list of $LR$-addresses, we can find an address that's not included.

Anyone have an idea how we could do this? How can we take any given list of $LR$-addresses and find one that's missing?

I'll give you a hint: This proof is called the "diagonal argument."

I'll be really impressed if anyone figures this out who hasn't seen it before. I didn't come up with this myself, someone had to show it to me!

Ok, here's how you do it.

Look at the first letter in the first address, and write down the opposite. Then look at the second letter in the second address, and write down the opposite. Keep doing this, diagonally, going down the list, forever and ever.

The new $LR$-address you just wrote down can't be anywhere in the list. Do you see why?

This new $LR$-address you just wrote down: Is it the first item in your rival's list?

No

no

no

no

no

can't be

Nope! The first letter is different.

no because the first letter is different

Exactly, it can't be, because they disagree on the first letter.

Is it the second item in the list?

no

no

no because the second letter is different

No, disagrees on the second letter

nope

no

no because second would be different

No, the second letter must be different

no because they disagree on the 2nd letter

Is it the $n$th item in the list?

NO

no, the nth letter would be different

No

no because they disagree in the nth letter

No, it disagrees on the nth letter

no

no they disagree on the nth letter

Nope!

No, the nth letter must be different

So it can't be anywhere in the list at all. It's different from every $LR$-address in this list.

This $LR$-address is missing from the list! Your rival was wrong — their list is incomplete.

haha take that rival

yay!

Take that, imaginary rival

What if they add it to their list

Good question. Anyone want to answer that? What do we do if they try to fix their list by adding the missing address we just found?

Do it again

then you do the same thing

we do the same thing again

then we can get a new one

do the same thing again

WANNA SEE ME DO IT AGAIN???

Just use the diagonal argument again.

Yeah, we can just run the same process again and find a new missing $LR$-address.

No matter what list they hand you, you can find a point that's missing! Even an infinite list of $LR$-addresses can't contain every $LR$-address.

So the total number of $LR$-addresses, aka the number of points on a line, really is bigger than infinity!

And what do we say at the end of a proof?

QED

QED

QED

QED!!!

Q.E.D

QED, boom

MIC DROP!

Q.E.D.

$QED$

QED

QED!

This is a super bizarre result: there's something bigger than infinity. Or, to be a little more precise, there are different sizes of infinity.

The thing that we've just been calling "infinity" is usually written as $\aleph_0$, pronounced "aleph-nought."

If you write it as $\infty$, like we've been doing today, people will be confused, because (as we just showed) there are different types of infinity.

$\aleph_0$ is the smallest infinity. It's the size of the set of all integers. You can type it like this: \aleph_0

And $\mathfrak{c}$, the continuum, is the size of the set of all real numbers. We just proved that $\mathfrak{c} > \aleph_0$.

So... this result raises some natural questions, right? What are you wondering about, now that you know that there are different sizes of infinity?

is there anything bigger than continuum?

is there an infinity between the two.

What's the greatest form of infinity?

How many infinities

*are*there?

are there even bigger infinities?

is there a biggest infinity?

all the different sizes

How many different sizes of infinity are there?

These are all great questions, and we don't have time to get to all of them. But I'll give you some quick answers.

Is there anything bigger than $\mathfrak{c}$? Yes, there is! In fact, there are infinitely many different sizes of infinity!

Is there anything between $\aleph_0$ and $\mathfrak{c}$?

This question is one of the weirdest questions in the history of math. It's called the "continuum hypothesis" if you want to look it up.

And the answer is... that there is no answer. It's impossible to prove there's something between $\aleph_0$ and $\mathfrak{c}$, and it's impossible to prove there

*isn't*something between $\aleph_0$ and $\mathfrak{c}$!

How is this proved?

how do we know it's impossible to prove?

wait... what????

How is it proved that this is impossible to prove?

that's a proven fact?

Yeah... it sounds ridiculous, but mathematicians have proven that it's impossible to prove the continuum hypothesis, true or false, using the standard axioms of math. Wild stuff!!

This is one of the most unsettling things about theoretical math. Some things can be proven true, some things can be proven false, and some things can be proven unprovable!

Anyway, that about wraps up what I had planned for us today. I hope everyone enjoyed this excursion into math without numbers.

If this kind of thing interests you, you should really check out my book Math Without Numbers. I explain a lot of my favorite math concepts: maps, dimensions, symmetry, abstract structures, manifolds, Conway's game of life, and even some stuff about the history and philosophy of math. (Did you know some people think the entire physical universe is a mathematical object?)

Plus there are a ton of little riddles and games and bonus materials thrown in, like how to draw a dodecahedron, or why you never see a full moon in the daytime.

You can order the book here: https://www.penguinrandomhouse.com/books/609844/math-without-numbers-by-milo-beckman/

I also adapted parts of the book into a little YouTube video series, which you can check out for free: https://www.youtube.com/channel/UCURP6q1VjGEFfy7GPP3Tm0w

That's all. Now that we're done, you can go ahead and type numbers again!

Yay! I love math books!

1+1=2

34567890987654323456789

12343

42

23

34

1

12

1290830194037598308925029384

Thanks everyone for participating!

And thank you Milo, for leading this fun discussion!

Bye!

THANK YOU

bye! thank you!

thx bye

Thank you!

Good night!

byebye

bye bye!

Thank you!