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Math Without Numbers - Infinity

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Is 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

vapodaca 2021-01-06 19:00:09
Hello and welcome to the Math Without Numbers Jam!
fangxcui 2021-01-06 19:00:22
cryptographer 2021-01-06 19:00:22
lasershark 2021-01-06 19:00:22
mathpower007 2021-01-06 19:00:22
Blueclay 2021-01-06 19:00:22
emma_maths 2021-01-06 19:00:22
leozhang 2021-01-06 19:00:22
OpalRaven 2021-01-06 19:00:22
vapodaca 2021-01-06 19:01:00
Now let me introduce our guest for the evening.
vapodaca 2021-01-06 19:01:07
Milo Beckman (miyomiyo) is an NYC Math Team alum and author of the newly-released book Math Without Numbers.
vapodaca 2021-01-06 19:01:15
Thanks for being here with us this evening, Milo!
miyomiyo 2021-01-06 19:01:21
Hey everyone! Welcome to a Math Jam... without numbers!
miyomiyo 2021-01-06 19:01:32
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.
miyomiyo 2021-01-06 19:01:49
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:
miyomiyo 2021-01-06 19:02:18
Today I want to give you a little preview of one of my favorite topics in abstract math, without using any numbers.
miyomiyo 2021-01-06 19:02:24
The topic is: infinity!
miyomiyo 2021-01-06 19:02:39
(Technically the topic is "infinite cardinalities and Cantor's diagonalization argument," but let's just say "infinity" for now.)
miyomiyo 2021-01-06 19:02:54
We're going to be focused on one big question: Is anything bigger than infinity?
miyomiyo 2021-01-06 19:03:02
Before we even get started, I want to hear your guesses. Just from your intuition, do you think there's something bigger than infinity?
Zanderman 2021-01-06 19:03:23
PenguinMoosey 2021-01-06 19:03:23
EandCheckmark 2021-01-06 19:03:23
RedFireTruck 2021-01-06 19:03:23
URcurious2 2021-01-06 19:03:23
RRao56 2021-01-06 19:03:23
yekolo 2021-01-06 19:03:23
dan09 2021-01-06 19:03:23
RedFireTruck 2021-01-06 19:03:23
blue_Yonder 2021-01-06 19:03:23
miyomiyo 2021-01-06 19:03:37
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.
miyomiyo 2021-01-06 19:03:51
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.
miyomiyo 2021-01-06 19:04:01
For our question "Is anything bigger than infinity?" - what words do you think we need to define?
yekolo 2021-01-06 19:04:24
Equinox8 2021-01-06 19:04:24
Math_Wiz13 2021-01-06 19:04:24
awesomediabrine 2021-01-06 19:04:24
STGpotter 2021-01-06 19:04:24
Equinox8 2021-01-06 19:04:24
EandCheckmark 2021-01-06 19:04:24
pandapo 2021-01-06 19:04:24
bigger and infinity
mathpower007 2021-01-06 19:04:24
Cubesat 2021-01-06 19:04:24
Infinity and bigger
miyomiyo 2021-01-06 19:04:31
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."
miyomiyo 2021-01-06 19:04:38
What does it mean when we say one thing is "bigger" than another?
miyomiyo 2021-01-06 19:04:57
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.
miyomiyo 2021-01-06 19:05:08
Let's come up with a rule—a totally solid, foolproof rule—that tells us when one quantity is "bigger" than another.
miyomiyo 2021-01-06 19:05:26
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?
miyomiyo 2021-01-06 19:05:28
miyomiyo 2021-01-06 19:05:44
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?
miyomiyo 2021-01-06 19:06:21
It's such a basic, obvious concept that it's actually pretty hard to describe from scratch.
miyomiyo 2021-01-06 19:06:36
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?
miyomiyo 2021-01-06 19:06:39
Equinox8 2021-01-06 19:07:19
they have an equal amount of objects
Zanderman 2021-01-06 19:07:19
They have the same amount of stuff
lasershark 2021-01-06 19:07:19
they have the same number of "things"
TheSubway 2021-01-06 19:07:19
They are identical in amounts of items.
miyomiyo 2021-01-06 19:07:24
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.
miyomiyo 2021-01-06 19:07:28
Any other ideas? Let's think about it for a minute.
Cubesat 2021-01-06 19:07:52
We can pair them off equally
snow_monkey 2021-01-06 19:07:52
you can pair up objects exactly with nothing left over on either side
awesomediabrine 2021-01-06 19:07:52
each item is able to be paired up
miyomiyo 2021-01-06 19:07:58
That's a great idea! We can tell the piles are equal because we can match them up perfectly without any leftovers.
miyomiyo 2021-01-06 19:08:10
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?
miyomiyo 2021-01-06 19:08:15
Look, I'll make it into a sticky:
miyomiyo 2021-01-06 19:08:19
Two piles are "equal" if you can match up their objects without any leftovers.
miyomiyo 2021-01-06 19:08:29
miyomiyo 2021-01-06 19:08:42
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?
Ladka13 2021-01-06 19:09:20
so if they aren't equal then after pairing the objects up one pile will have some leftovers
RedFireTruck 2021-01-06 19:09:20
if the pile has leftovers then it is bigger
ThinkingThings 2021-01-06 19:09:20
After matching the objects up, the group with leftovers is bigger than the other one.
TheSubway 2021-01-06 19:09:20
When there is a remainder after pairing.
miyomiyo 2021-01-06 19:09:29
There we go! That's a great definition for the word "bigger." Let's sticky that too.
miyomiyo 2021-01-06 19:09:35
If you can't match up two piles perfectly, the side with leftovers is the "bigger" pile.
miyomiyo 2021-01-06 19:09:47
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?
SparklyFlowers 2021-01-06 19:10:04
Blossom1 2021-01-06 19:10:04
SundayMorning 2021-01-06 19:10:04
megahertz13 2021-01-06 19:10:04
PackingPeanuts 2021-01-06 19:10:04
STGpotter 2021-01-06 19:10:04
lasershark 2021-01-06 19:10:04
TheSubway 2021-01-06 19:10:04
Equinox8 2021-01-06 19:10:04
miyomiyo 2021-01-06 19:10:19
I'm still not going to tell you the answer just yet. We have to figure it out.
miyomiyo 2021-01-06 19:10:37
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.
miyomiyo 2021-01-06 19:10:48
The technical definition is: It's the size of the set of all natural numbers.
miyomiyo 2021-01-06 19:10:54
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.
miyomiyo 2021-01-06 19:10:57
miyomiyo 2021-01-06 19:11:15
How could anything be bigger than that?
s214425 2021-01-06 19:11:43
infinity + 1?
RickMark 2021-01-06 19:11:43
TomBradygoat 2021-01-06 19:11:43
+ 1 more?
miyomiyo 2021-01-06 19:11:48
Yeah, what about infinity plus one?
miyomiyo 2021-01-06 19:11:53
miyomiyo 2021-01-06 19:12:00
It doesn't seem like one extra object should make any difference, but let's use our "bigger" rule just to make sure.
miyomiyo 2021-01-06 19:12:09
First I'm going to arrange our objects in a line so they're easier to keep track of.
miyomiyo 2021-01-06 19:12:16
miyomiyo 2021-01-06 19:12:26
If we try to match up the objects in the obvious way, it certainly looks like infinity plus one is bigger...
miyomiyo 2021-01-06 19:12:30
miyomiyo 2021-01-06 19:12:43
But is there a way to match them up without leftovers?
dan09 2021-01-06 19:13:15
You can shift infinity to the right to match up
Cubesat 2021-01-06 19:13:15
Yes! match up the extra first
ianlee25510 2021-01-06 19:13:15
if you scooch one back?
miyomiyo 2021-01-06 19:13:20
Yep, there is. Check it out:
miyomiyo 2021-01-06 19:13:23
miyomiyo 2021-01-06 19:13:36
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.
miyomiyo 2021-01-06 19:13:59
So the two piles are the same size! Infinity plus one equals infinity.
miyomiyo 2021-01-06 19:14:05
That's pretty bizarre, right?
OpalRaven 2021-01-06 19:14:23
SparklyFlowers 2021-01-06 19:14:23
what the
adatta0517 2021-01-06 19:14:23
miyomiyo 2021-01-06 19:14:31
Here's another way of looking at the same problem. This is a famous example, called Hilbert's Hotel.
miyomiyo 2021-01-06 19:14:39
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.
miyomiyo 2021-01-06 19:14:56
miyomiyo 2021-01-06 19:15:10
Tonight is an especially busy night: Every room in the hotel is full. (Yes, this world has infinity people, too.)
miyomiyo 2021-01-06 19:15:20
Then someone walks into the hotel lobby from the outside world and says, “Could I please have a room?”
miyomiyo 2021-01-06 19:15:29
What do you do? Do you tell them the hotel is full and turn them away?
ARSM2019 2021-01-06 19:15:53
Just move everyone to the right one room
timchen 2021-01-06 19:15:53
shift everyone up by one
yitesr 2021-01-06 19:15:53
everyone moves to the next room
Equinox8 2021-01-06 19:15:53
move everyone down to room number + 1
Ron.Livne 2021-01-06 19:15:53
you would ask them all to move to n + 1 room
lasershark 2021-01-06 19:15:53
do you move everybody over 1
miyomiyo 2021-01-06 19:16:06
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.”
miyomiyo 2021-01-06 19:16:24
Once everyone does what you say, you’ve cleared out a room for the new guest.
miyomiyo 2021-01-06 19:16:27
miyomiyo 2021-01-06 19:17:10
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.
miyomiyo 2021-01-06 19:17:29
Ok, so we're all agreed: infinity plus one is still infinity.
miyomiyo 2021-01-06 19:17:32
What about, I don't know, infinity plus five?
sushi1233 2021-01-06 19:17:49
it's still infiinty
pyz18 2021-01-06 19:17:49
Ladka13 2021-01-06 19:17:49
same thing
SparklyFlowers 2021-01-06 19:17:49
equals infinity
Zerubbabel 2021-01-06 19:17:49
still just infinity
Blossom1 2021-01-06 19:17:49
yep! still equal
mathKidMom 2021-01-06 19:17:49
still infinity
miyomiyo 2021-01-06 19:17:57
Yeah— it doesn't make a difference. We could use the exact same trick: just have everyone move down five doors instead.
miyomiyo 2021-01-06 19:18:08
Even infinity plus a billion is still just infinity. Any finite number added to infinity, you can match it up to infinity without leftovers.
miyomiyo 2021-01-06 19:18:21
What else could we try? What might be bigger than infinity?
twsaving2 2021-01-06 19:18:39
What is infinity+infinity then?
dan09 2021-01-06 19:18:39
Infinity plus infinity?
Zanderman 2021-01-06 19:18:39
STGpotter 2021-01-06 19:18:39
what about infinity + infinity
erm999 2021-01-06 19:18:39
infinty + infinity???
snow_monkey 2021-01-06 19:18:39
jessieH2 2021-01-06 19:18:39
but what about infinity plus infinity
HappyLemon07 2021-01-06 19:18:39
what about infinity plus infinity?
sushi1233 2021-01-06 19:18:39
infinity + infinity
miyomiyo 2021-01-06 19:18:50
There's an idea: What about infinity plus infinity?
miyomiyo 2021-01-06 19:18:55
Can two infinity bags be matched up with one?
miyomiyo 2021-01-06 19:18:59
miyomiyo 2021-01-06 19:19:17
Or, in the hotel example, can we find rooms for an infinite line of new guests?
miyomiyo 2021-01-06 19:19:23
miyomiyo 2021-01-06 19:19:55
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?
miyomiyo 2021-01-06 19:20:23
So is it impossible? What do you think?
Tong.Qiu 2021-01-06 19:20:48
we could tell them to go to room 2n
awesomeming327. 2021-01-06 19:20:48
1 goes to 2, 2 goes to 4, 3 goes to 6
Ron.Livne 2021-01-06 19:20:48
they move to their room number times 2
NonOxyCrustacean 2021-01-06 19:20:48
room one goes to room two, room 2 goes to room 4, etc etc
crosby88 2021-01-06 19:20:48
put people in their room number * 2
miyomiyo 2021-01-06 19:21:00
Exactly! The key is to space them out.
miyomiyo 2021-01-06 19:21:08
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 $2n$?"
miyomiyo 2021-01-06 19:21:12
miyomiyo 2021-01-06 19:21:26
(Darn, I used a number. I promise it won't happen again.)
miyomiyo 2021-01-06 19:21:42
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.
miyomiyo 2021-01-06 19:22:01
Here's the same argument with bags of objects:
miyomiyo 2021-01-06 19:22:04
miyomiyo 2021-01-06 19:22:34
So there you have it: Infinity plus infinity equals infinity.
miyomiyo 2021-01-06 19:22:50
Are we ready to give up? Or does anyone think there's something bigger?
Equinox8 2021-01-06 19:23:12
what about infinity times infinity
Fatsteak 2021-01-06 19:23:12
goodskate 2021-01-06 19:23:12
what about infinity * infinity though?
TheSubway 2021-01-06 19:23:12
Can we try $\infty \times \infty$?
Mathdreams 2021-01-06 19:23:12
Infinity times infinity!
NonOxyCrustacean 2021-01-06 19:23:12
what about infinity*infinity?
TheSubway 2021-01-06 19:23:12
Inifnity times infinity.
miyomiyo 2021-01-06 19:23:20
Ok, what about infinity TIMES infinity?
miyomiyo 2021-01-06 19:23:28
So... something that looks like this:
miyomiyo 2021-01-06 19:23:31
miyomiyo 2021-01-06 19:23:51
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?
SparklyFlowers 2021-01-06 19:24:16
pandapo 2021-01-06 19:24:16
Equinox8 2021-01-06 19:24:16
URcurious2 2021-01-06 19:24:16
GalaxyDragon09 2021-01-06 19:24:16
megahertz13 2021-01-06 19:24:16
sushi1233 2021-01-06 19:24:16
still equal
Cubesat 2021-01-06 19:24:16
Mathfan9 2021-01-06 19:24:16
Zanderman 2021-01-06 19:24:16
Tong.Qiu 2021-01-06 19:24:16
goodskate 2021-01-06 19:24:16
Mathdreams 2021-01-06 19:24:16
BlueyStudiosYT 2021-01-06 19:24:16
bdcl 2021-01-06 19:24:16
Hornet 2021-01-06 19:24:16
still equal
BananaChamp1 2021-01-06 19:24:16
scube20 2021-01-06 19:24:16
Pinepiano 2021-01-06 19:24:16
miyomiyo 2021-01-06 19:24:30
Can anyone prove their answer is right? Guessing is one thing, but I want to see some proof.
miyomiyo 2021-01-06 19:24:38
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.
miyomiyo 2021-01-06 19:24:50
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.
miyomiyo 2021-01-06 19:25:09
Any ideas? I'll give you a minute to think about it.
miyomiyo 2021-01-06 19:25:31
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?
miyomiyo 2021-01-06 19:26:59
The answer is EQUAL. Infinity times infinity still equals infinity.
miyomiyo 2021-01-06 19:27:05
Want to see a proof?
NG2021 2021-01-06 19:27:18
rench23sugar 2021-01-06 19:27:18
puppyjohnson 2021-01-06 19:27:18
Mathfan9 2021-01-06 19:27:18
yes please
ploik11 2021-01-06 19:27:18
pinkpig 2021-01-06 19:27:18
miyomiyo 2021-01-06 19:27:22
This is called the "snake proof" and I think it's pretty neat.
miyomiyo 2021-01-06 19:27:26
miyomiyo 2021-01-06 19:27:58
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.
miyomiyo 2021-01-06 19:28:39
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.
miyomiyo 2021-01-06 19:28:47
("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.")
miyomiyo 2021-01-06 19:29:05
So there you go, QED. Does everyone find this proof convincing?
jfelicita 2021-01-06 19:29:25
ianlee25510 2021-01-06 19:29:25
Cool times infinity!
firebolt360 2021-01-06 19:29:25
Oh thats pretty neat!
edgymemelord 2021-01-06 19:29:25
JaiT 2021-01-06 19:29:25
STGpotter 2021-01-06 19:29:25
emma_maths 2021-01-06 19:29:25
amalgam1971 2021-01-06 19:29:25
brendan_cape 2021-01-06 19:29:25
miyomiyo 2021-01-06 19:29:38
Great, so it's settled. Infinity times infinity equals infinity.
miyomiyo 2021-01-06 19:29:42
Are we ready to give up? What do you all think: Is there something bigger than infinity?
ianlee25510 2021-01-06 19:30:22
KT22 2021-01-06 19:30:22
yitesr 2021-01-06 19:30:22
neev12 2021-01-06 19:30:22
Joshua_Oh 2021-01-06 19:30:22
No there is nothing bigger than infinity
iqzhang 2021-01-06 19:30:22
pinkpig 2021-01-06 19:30:22
Nothing is bigger.
JaiT 2021-01-06 19:30:22
OpalRaven 2021-01-06 19:30:22
Cubesat 2021-01-06 19:30:22
Tong.Qiu 2021-01-06 19:30:22
I'm pretty sure there is something bigger
NG2021 2021-01-06 19:30:22
nope, there is nothing bigger than infinity
miyomiyo 2021-01-06 19:30:34
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?
ianlee25510 2021-01-06 19:30:54
aops-g5-gethsemanea2 2021-01-06 19:30:54
pinkpig 2021-01-06 19:30:54
mathKidMom 2021-01-06 19:30:54
RRao56 2021-01-06 19:30:54
miyomiyo 2021-01-06 19:31:04
There IS something bigger than infinity!
miyomiyo 2021-01-06 19:31:20
It's called "the continuum."
miyomiyo 2021-01-06 19:31:28
We usually write it with a fancy lowercase c, like this: $\mathfrak{c}$
miyomiyo 2021-01-06 19:31:35
You can type it like this: \mathfrak{c}.
miyomiyo 2021-01-06 19:32:03
So what I'm claiming now, and what we're about to try to prove, is this: $\mathfrak{c} > \infty$.
miyomiyo 2021-01-06 19:32:12
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?
miyomiyo 2021-01-06 19:32:24
There are a lot of ways to define $\mathfrak{c}$, but here's my favorite: it's the number of points in a line.
miyomiyo 2021-01-06 19:32:40
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.
miyomiyo 2021-01-06 19:32:51
Let's look at this on the number line.
miyomiyo 2021-01-06 19:32:54
miyomiyo 2021-01-06 19:33:10
Which of these pictures represents $\infty$ and which represents $\mathfrak{c}$?
sosiaops 2021-01-06 19:33:43
c is on top
Ron.Livne 2021-01-06 19:33:43
the top one is c
turtle22 2021-01-06 19:33:43
bottom is infinity
STGpotter 2021-01-06 19:33:43
c represents the first one
awesomeness07 2021-01-06 19:33:43
top is c and bottom is infinity
miyomiyo 2021-01-06 19:33:51
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!
miyomiyo 2021-01-06 19:34:12
So, let's check if this is making sense. How many positive integers are there? $\infty$ or $\mathfrak{c}$?
dan09 2021-01-06 19:34:29
Blossom1 2021-01-06 19:34:29
discula2020 2021-01-06 19:34:29
goodskate 2021-01-06 19:34:29
MLiang2018 2021-01-06 19:34:29
ThinkingThings 2021-01-06 19:34:29
Tong.Qiu 2021-01-06 19:34:29
pyz18 2021-01-06 19:34:29
miyomiyo 2021-01-06 19:34:46
Yep, $\infty$. And how many positive integers are there total — positive, negative, and zero? $\infty$ or $\mathfrak{c}$?
edgymemelord 2021-01-06 19:35:21
still infinity
TheSubway 2021-01-06 19:35:21
NonOxyCrustacean 2021-01-06 19:35:21
EandCheckmark 2021-01-06 19:35:21
SparklyFlowers 2021-01-06 19:35:21
still infinity
mathKidMom 2021-01-06 19:35:21
prickly-pear 2021-01-06 19:35:21
Tong.Qiu 2021-01-06 19:35:21
infinity still
miyomiyo 2021-01-06 19:35:27
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, $\infty + \infty = \infty$.
miyomiyo 2021-01-06 19:35:44
Ok, how many REAL numbers are there? Including fractions, square roots, transcendental numbers like $\pi$ and $e$, and everything in between?
lnzhonglp 2021-01-06 19:36:19
TheSubway 2021-01-06 19:36:19
Tong.Qiu 2021-01-06 19:36:19
now it's c
snow_monkey 2021-01-06 19:36:19
Quentissential 2021-01-06 19:36:19
UnknownMonkey 2021-01-06 19:36:19
duo_duo 2021-01-06 19:36:19
GabeW 2021-01-06 19:36:19
Cubesat 2021-01-06 19:36:19
HappyLemon07 2021-01-06 19:36:19
the fancy c
miyomiyo 2021-01-06 19:36:26
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.
miyomiyo 2021-01-06 19:36:56
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.
miyomiyo 2021-01-06 19:37:21
Now how on Earth are we going to prove something like that? Let's check back on our definitions in those stickies up top.
miyomiyo 2021-01-06 19:37:33
What do we need to do in order to prove that the continuum is "bigger" than infinity?
NonOxyCrustacean 2021-01-06 19:38:26
it has leftovers when compared to infinity
TheSubway 2021-01-06 19:38:26
That there are leftovers after pairing.
ThinkingThings 2021-01-06 19:38:26
See if there's leftovers
snow_monkey 2021-01-06 19:38:26
theres leftovers after pairing
scube20 2021-01-06 19:38:26
making sure there is left over on the c side
miyomiyo 2021-01-06 19:38:51
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.
miyomiyo 2021-01-06 19:39:26
We need to prove that it's impossible to match up the points on a line with the objects in a bottomless bag.
miyomiyo 2021-01-06 19:39:37
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.
miyomiyo 2021-01-06 19:40:05
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?
miyomiyo 2021-01-06 19:40:47
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.
TheSubway 2021-01-06 19:41:28
NG2021 2021-01-06 19:41:28
LightningStar 2021-01-06 19:41:28
Elektrolyte 2021-01-06 19:41:28
SparklyFlowers 2021-01-06 19:41:28
fangxcui 2021-01-06 19:41:28
masadca 2021-01-06 19:41:28
miyomiyo 2021-01-06 19:41:43
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!)
miyomiyo 2021-01-06 19:42:12
We're going to assign every point on the line segment an "$LR$-address" in a neat, systematic way.
miyomiyo 2021-01-06 19:42:25
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.
miyomiyo 2021-01-06 19:42:41
The second letter tells you if the point is on the left half ($L$) or right half ($R$) of that half.
miyomiyo 2021-01-06 19:42:52
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.
jfelicita 2021-01-06 19:43:08
But, what if its in the middle
Cubesat 2021-01-06 19:43:08
What if it's in the middle?
miyomiyo 2021-01-06 19:43:19
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.
miyomiyo 2021-01-06 19:43:33
Here's an example of an arbitrary point on the line segment, and how we would start to give it an $LR$-address.
miyomiyo 2021-01-06 19:43:36
miyomiyo 2021-01-06 19:44:33
Can someone tell me what $LR$-address we would give to the point all the way on the left of the segment?
josh924 2021-01-06 19:45:20
Cubesat 2021-01-06 19:45:20
dan09 2021-01-06 19:45:20
LL... to infinity
Mathdreams 2021-01-06 19:45:20
Ariva 2021-01-06 19:45:20
StickyWashington 2021-01-06 19:45:20
LL... repeating forever
miyomiyo 2021-01-06 19:45:30
Yep: that point is named $LLLLLLLL\ldots$ (continuing forever).
miyomiyo 2021-01-06 19:45:50
What about the point that's three-quarters of the way to the right?
yekolo 2021-01-06 19:46:22
URcurious2 2021-01-06 19:46:22
TheSubway 2021-01-06 19:46:22
ChiaravalleLearner 2021-01-06 19:46:22
edgymemelord 2021-01-06 19:46:22
JCJC 2021-01-06 19:46:22
Equinox8 2021-01-06 19:46:22
miyomiyo 2021-01-06 19:46:46
That one's just named $RM$. You go to the right half, and then your point is exactly in the middle.
miyomiyo 2021-01-06 19:47:12
And what about the point that's one-third of the way across? Think about it.
miyomiyo 2021-01-06 19:47:26
(This one's pretty tricky.)
boing123 2021-01-06 19:48:23
yekolo 2021-01-06 19:48:23
JazzyTheJazzer1234 2021-01-06 19:48:23
ARSM2019 2021-01-06 19:48:23
Cubesat 2021-01-06 19:48:23
Mathdreams 2021-01-06 19:48:23
speedstar 2021-01-06 19:48:23
goodskate 2021-01-06 19:48:23
LRLRLRLRLRLR...repeating forever
NonOxyCrustacean 2021-01-06 19:48:23
miyomiyo 2021-01-06 19:48:50
That one would be named $LRLRLRLR\ldots$ (repeating forever).
miyomiyo 2021-01-06 19:48:55
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.
miyomiyo 2021-01-06 19:49:15
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.
miyomiyo 2021-01-06 19:49:37
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.
miyomiyo 2021-01-06 19:50:18
miyomiyo 2021-01-06 19:50:39
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.
miyomiyo 2021-01-06 19:51:08
How can we prove their list is incomplete? Any ideas?
miyomiyo 2021-01-06 19:51:43
What's the least we could do to prove that this proposed list is incomplete?
edgymemelord 2021-01-06 19:52:31
make a new address
timchen 2021-01-06 19:52:31
find something that is missing
URcurious2 2021-01-06 19:52:31
find a point not on the list
Mathfan9 2021-01-06 19:52:31
get an LR addres that can't be on the list
miyomiyo 2021-01-06 19:52:43
Right, we just need to find one missing $LR$-address. That's enough to show they were wrong.
miyomiyo 2021-01-06 19:53:00
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.
miyomiyo 2021-01-06 19:53:49
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?
miyomiyo 2021-01-06 19:54:22
I'll give you a hint: This proof is called the "diagonal argument."
miyomiyo 2021-01-06 19:54:37
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!
miyomiyo 2021-01-06 19:55:59
Ok, here's how you do it.
miyomiyo 2021-01-06 19:56:10
miyomiyo 2021-01-06 19:56:15
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.
miyomiyo 2021-01-06 19:57:04
The new $LR$-address you just wrote down can't be anywhere in the list. Do you see why?
miyomiyo 2021-01-06 19:57:50
This new $LR$-address you just wrote down: Is it the first item in your rival's list?
pooja26678 2021-01-06 19:58:16
ARSM2019 2021-01-06 19:58:16
Mathfan9 2021-01-06 19:58:16
mathKidMom 2021-01-06 19:58:16
JazzyTheJazzer1234 2021-01-06 19:58:16
prickly-pear 2021-01-06 19:58:16
can't be
Cubesat 2021-01-06 19:58:16
Nope! The first letter is different.
bapmookja 2021-01-06 19:58:16
no because the first letter is different
miyomiyo 2021-01-06 19:58:27
Exactly, it can't be, because they disagree on the first letter.
miyomiyo 2021-01-06 19:58:32
Is it the second item in the list?
pooja26678 2021-01-06 19:58:59
cutesamoyeds 2021-01-06 19:58:59
bapmookja 2021-01-06 19:58:59
no because the second letter is different
Quentissential 2021-01-06 19:58:59
No, disagrees on the second letter
mathKidMom 2021-01-06 19:58:59
t_ameya 2021-01-06 19:58:59
goodskate 2021-01-06 19:58:59
no because second would be different
krishgarg 2021-01-06 19:58:59
No, the second letter must be different
Equinox8 2021-01-06 19:58:59
no because they disagree on the 2nd letter
miyomiyo 2021-01-06 19:59:10
Is it the $n$th item in the list?
pooja26678 2021-01-06 19:59:39
Mathfan9 2021-01-06 19:59:39
no, the nth letter would be different
URcurious2 2021-01-06 19:59:39
mt2021 2021-01-06 19:59:39
no because they disagree in the nth letter
Quentissential 2021-01-06 19:59:39
No, it disagrees on the nth letter
mathKidMom 2021-01-06 19:59:39
RRao56 2021-01-06 19:59:39
no they disagree on the nth letter
Cubesat 2021-01-06 19:59:39
krishgarg 2021-01-06 19:59:39
No, the nth letter must be different
miyomiyo 2021-01-06 20:00:09
So it can't be anywhere in the list at all. It's different from every $LR$-address in this list.
miyomiyo 2021-01-06 20:00:32
This $LR$-address is missing from the list! Your rival was wrong — their list is incomplete.
Equinox8 2021-01-06 20:01:09
haha take that rival
mathKidMom 2021-01-06 20:01:09
jfelicita 2021-01-06 20:01:09
Take that, imaginary rival
RRao56 2021-01-06 20:01:23
What if they add it to their list
miyomiyo 2021-01-06 20:01:30
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?
jfelicita 2021-01-06 20:02:02
Do it again
HappyLemon07 2021-01-06 20:02:02
then you do the same thing
mt2021 2021-01-06 20:02:02
we do the same thing again
ColtsFan10 2021-01-06 20:02:02
then we can get a new one
erm999 2021-01-06 20:02:02
do the same thing again
EandCheckmark 2021-01-06 20:02:02
Lily55 2021-01-06 20:02:02
Just use the diagonal argument again.
miyomiyo 2021-01-06 20:02:16
Yeah, we can just run the same process again and find a new missing $LR$-address.
miyomiyo 2021-01-06 20:02:43
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.
miyomiyo 2021-01-06 20:03:26
So the total number of $LR$-addresses, aka the number of points on a line, really is bigger than infinity!
miyomiyo 2021-01-06 20:03:45
And what do we say at the end of a proof?
goodskate 2021-01-06 20:04:11
pyz18 2021-01-06 20:04:11
blue_Yonder 2021-01-06 20:04:11
jfelicita 2021-01-06 20:04:11
Mathfan9 2021-01-06 20:04:11
pyz18 2021-01-06 20:04:11
QED, boom
Korra 2021-01-06 20:04:11
Quentissential 2021-01-06 20:04:11
URcurious2 2021-01-06 20:04:11
prickly-pear 2021-01-06 20:04:11
miyomiyo 2021-01-06 20:04:18
miyomiyo 2021-01-06 20:04:31
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.
miyomiyo 2021-01-06 20:04:54
The thing that we've just been calling "infinity" is usually written as $\aleph_0$, pronounced "aleph-nought."
miyomiyo 2021-01-06 20:05:17
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.
miyomiyo 2021-01-06 20:05:35
$\aleph_0$ is the smallest infinity. It's the size of the set of all integers. You can type it like this: \aleph_0
miyomiyo 2021-01-06 20:06:49
And $\mathfrak{c}$, the continuum, is the size of the set of all real numbers. We just proved that $\mathfrak{c} > \aleph_0$.
miyomiyo 2021-01-06 20:07:18
So... this result raises some natural questions, right? What are you wondering about, now that you know that there are different sizes of infinity?
Mathfan9 2021-01-06 20:08:54
is there anything bigger than continuum?
Ningster 2021-01-06 20:08:54
is there an infinity between the two.
Elektrolyte 2021-01-06 20:08:54
What's the greatest form of infinity?
yekolo 2021-01-06 20:08:54
How many infinities are there?
ARSM2019 2021-01-06 20:08:54
are there even bigger infinities?
prickly-pear 2021-01-06 20:08:54
is there a biggest infinity?
neev12 2021-01-06 20:08:54
all the different sizes
jfelicita 2021-01-06 20:08:54
How many different sizes of infinity are there?
miyomiyo 2021-01-06 20:09:15
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.
miyomiyo 2021-01-06 20:09:33
Is there anything bigger than $\mathfrak{c}$? Yes, there is! In fact, there are infinitely many different sizes of infinity!
miyomiyo 2021-01-06 20:10:07
Is there anything between $\aleph_0$ and $\mathfrak{c}$?
miyomiyo 2021-01-06 20:10:16
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.
miyomiyo 2021-01-06 20:10:46
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 $\mathfrak{c}$!
yekolo 2021-01-06 20:11:30
How is this proved?
ab2024 2021-01-06 20:11:30
how do we know it's impossible to prove?
goodskate 2021-01-06 20:11:30
wait... what????
yekolo 2021-01-06 20:11:30
How is it proved that this is impossible to prove?
snow_monkey 2021-01-06 20:11:30
that's a proven fact?
miyomiyo 2021-01-06 20:12:27
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!!
miyomiyo 2021-01-06 20:13:08
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!
miyomiyo 2021-01-06 20:14:16
Anyway, that about wraps up what I had planned for us today. I hope everyone enjoyed this excursion into math without numbers.
miyomiyo 2021-01-06 20:14:33
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?)
miyomiyo 2021-01-06 20:15:02
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.
miyomiyo 2021-01-06 20:15:17
You can order the book here:
miyomiyo 2021-01-06 20:15:26
I also adapted parts of the book into a little YouTube video series, which you can check out for free:
miyomiyo 2021-01-06 20:15:42
That's all. Now that we're done, you can go ahead and type numbers again!
Cubesat 2021-01-06 20:16:18
Yay! I love math books!
ThinkingThings 2021-01-06 20:16:18
adatta0517 2021-01-06 20:16:18
kpatel0581 2021-01-06 20:16:18
bronzetruck2016 2021-01-06 20:16:18
mjjiang 2021-01-06 20:16:18
Equinox8 2021-01-06 20:16:18
URcurious2 2021-01-06 20:16:18
URcurious2 2021-01-06 20:16:18
mathKidMom 2021-01-06 20:16:18
vapodaca 2021-01-06 20:18:52
Thanks everyone for participating!
vapodaca 2021-01-06 20:19:14
And thank you Milo, for leading this fun discussion!
Maxlovesairandteslas1 2021-01-06 20:21:16
Pikachu19 2021-01-06 20:21:16
Blossom1 2021-01-06 20:21:16
bye! thank you!
Epicsmartypants 2021-01-06 20:21:16
thx bye
superm8 2021-01-06 20:21:16
Thank you!
hashtagmath 2021-01-06 20:21:16
Good night!
Maxlovesairandteslas1 2021-01-06 20:21:16
zby 2021-01-06 20:21:16
bye bye!
Maxlovesairandteslas1 2021-01-06 20:21:16
Thank you!

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