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
vapodaca
2021-01-06 19:00:09
Hello and welcome to the Math Without Numbers Jam!
Hello and welcome to the Math Without Numbers Jam!
fangxcui
2021-01-06 19:00:22
hi
hi
cryptographer
2021-01-06 19:00:22
hi
hi
lasershark
2021-01-06 19:00:22
hi
hi
mathpower007
2021-01-06 19:00:22
hi
hi
Blueclay
2021-01-06 19:00:22
Salve!
Salve!
emma_maths
2021-01-06 19:00:22
HI!!
HI!!
leozhang
2021-01-06 19:00:22
hi
hi
OpalRaven
2021-01-06 19:00:22
yay!
yay!
vapodaca
2021-01-06 19:01:00
Now let me introduce our guest for the evening.
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.
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!
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!
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.
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: https://www.penguinrandomhouse.com/books/609844/math-without-numbers-by-milo-beckman/
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/
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.
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!
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.)
(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?
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?
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
Nope
Nope
PenguinMoosey
2021-01-06 19:03:23
nope
nope
EandCheckmark
2021-01-06 19:03:23
yes
yes
RedFireTruck
2021-01-06 19:03:23
NOPE!
NOPE!
URcurious2
2021-01-06 19:03:23
Yes
Yes
RRao56
2021-01-06 19:03:23
Hmm...
Hmm...
yekolo
2021-01-06 19:03:23
No
No
dan09
2021-01-06 19:03:23
No
No
RedFireTruck
2021-01-06 19:03:23
NOPE
NOPE
blue_Yonder
2021-01-06 19:03:23
yes
yes
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.
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.
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?
For our question "Is anything bigger than infinity?" - what words do you think we need to define?
yekolo
2021-01-06 19:04:24
Infinity
Infinity
Equinox8
2021-01-06 19:04:24
"Infinity"
"Infinity"
Math_Wiz13
2021-01-06 19:04:24
infinity
infinity
awesomediabrine
2021-01-06 19:04:24
infinity
infinity
STGpotter
2021-01-06 19:04:24
infinity
infinity
Equinox8
2021-01-06 19:04:24
"Bigger"
"Bigger"
EandCheckmark
2021-01-06 19:04:24
infinity
infinity
pandapo
2021-01-06 19:04:24
bigger and infinity
bigger and infinity
mathpower007
2021-01-06 19:04:24
infinity
infinity
Cubesat
2021-01-06 19:04:24
Infinity and bigger
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."
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?
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.
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.
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?
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?
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.
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?
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
they have an equal amount of objects
Zanderman
2021-01-06 19:07:19
They have the same amount of stuff
They have the same amount of stuff
lasershark
2021-01-06 19:07:19
they have the same number of "things"
they have the same number of "things"
TheSubway
2021-01-06 19:07:19
They are identical in amounts of items.
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.
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.
Any other ideas? Let's think about it for a minute.
Cubesat
2021-01-06 19:07:52
We can pair them off equally
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
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
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.
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?
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:
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.
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?
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
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
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.
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.
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.
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.
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?
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
No
No
Blossom1
2021-01-06 19:10:04
No
No
SundayMorning
2021-01-06 19:10:04
yes
yes
megahertz13
2021-01-06 19:10:04
no
no
PackingPeanuts
2021-01-06 19:10:04
no
no
STGpotter
2021-01-06 19:10:04
Nope
Nope
lasershark
2021-01-06 19:10:04
yes
yes
TheSubway
2021-01-06 19:10:04
Yes.
Yes.
Equinox8
2021-01-06 19:10:04
no
no
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.
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.
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.
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.
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?
How could anything be bigger than that?
s214425
2021-01-06 19:11:43
infinity + 1?
infinity + 1?
RickMark
2021-01-06 19:11:43
+1
+1
TomBradygoat
2021-01-06 19:11:43
+ 1 more?
+ 1 more?
miyomiyo
2021-01-06 19:11:48
Yeah, what about infinity plus one?
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.
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.
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...
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?
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
You can shift infinity to the right to match up
Cubesat
2021-01-06 19:13:15
Yes! match up the extra first
Yes! match up the extra first
ianlee25510
2021-01-06 19:13:15
if you scooch one back?
if you scooch one back?
miyomiyo
2021-01-06 19:13:20
Yep, there is. Check it out:
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.
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.
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?
That's pretty bizarre, right?
OpalRaven
2021-01-06 19:14:23
SparklyFlowers
2021-01-06 19:14:23
what the
what the
adatta0517
2021-01-06 19:14:23
Cool
Cool
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.
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.
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.)
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?”
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?
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
Just move everyone to the right one room
timchen
2021-01-06 19:15:53
shift everyone up by one
shift everyone up by one
yitesr
2021-01-06 19:15:53
everyone moves to the next room
everyone moves to the next room
Equinox8
2021-01-06 19:15:53
move everyone down to room number + 1
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
you would ask them all to move to n + 1 room
lasershark
2021-01-06 19:15:53
do you move everybody over 1
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.”
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.
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.
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.
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?
What about, I don't know, infinity plus five?
sushi1233
2021-01-06 19:17:49
it's still infiinty
it's still infiinty
pyz18
2021-01-06 19:17:49
infinity
infinity
Ladka13
2021-01-06 19:17:49
same thing
same thing
SparklyFlowers
2021-01-06 19:17:49
equals infinity
equals infinity
Zerubbabel
2021-01-06 19:17:49
still just infinity
still just infinity
Blossom1
2021-01-06 19:17:49
yep! still equal
yep! still equal
mathKidMom
2021-01-06 19:17:49
still infinity
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.
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.
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?
What else could we try? What might be bigger than infinity?
twsaving2
2021-01-06 19:18:39
What is infinity+infinity then?
What is infinity+infinity then?
dan09
2021-01-06 19:18:39
Infinity plus infinity?
Infinity plus infinity?
Zanderman
2021-01-06 19:18:39
infinity+infinity?
infinity+infinity?
STGpotter
2021-01-06 19:18:39
what about infinity + infinity
what about infinity + infinity
erm999
2021-01-06 19:18:39
infinty + infinity???
infinty + infinity???
snow_monkey
2021-01-06 19:18:39
infinity+infinity?
infinity+infinity?
jessieH2
2021-01-06 19:18:39
but what about infinity plus infinity
but what about infinity plus infinity
HappyLemon07
2021-01-06 19:18:39
what about infinity plus infinity?
what about infinity plus infinity?
sushi1233
2021-01-06 19:18:39
infinity + infinity
infinity + infinity
miyomiyo
2021-01-06 19:18:50
There's an idea: What about infinity plus infinity?
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?
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?
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?
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?
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
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
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
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
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
put people in their room number * 2
miyomiyo
2021-01-06 19:21:00
Exactly! The key is to space them out.
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 $2n$?"
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.)
(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.
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:
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.
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?
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
what about infinity times infinity
Fatsteak
2021-01-06 19:23:12
NOW FOR INFINITY TIMES INFINITY
NOW FOR INFINITY TIMES INFINITY
goodskate
2021-01-06 19:23:12
what about infinity * infinity though?
what about infinity * infinity though?
TheSubway
2021-01-06 19:23:12
Can we try $\infty \times \infty$?
Can we try $\infty \times \infty$?
Mathdreams
2021-01-06 19:23:12
Infinity times infinity!
Infinity times infinity!
NonOxyCrustacean
2021-01-06 19:23:12
what about infinity*infinity?
what about infinity*infinity?
TheSubway
2021-01-06 19:23:12
Inifnity times infinity.
Inifnity times infinity.
miyomiyo
2021-01-06 19:23:20
Ok, what about infinity TIMES infinity?
Ok, what about infinity TIMES infinity?
miyomiyo
2021-01-06 19:23:28
So... something that looks like this:
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?
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
equal
equal
pandapo
2021-01-06 19:24:16
equal
equal
Equinox8
2021-01-06 19:24:16
equal
equal
URcurious2
2021-01-06 19:24:16
equal
equal
GalaxyDragon09
2021-01-06 19:24:16
hmm
hmm
megahertz13
2021-01-06 19:24:16
equal
equal
sushi1233
2021-01-06 19:24:16
still equal
still equal
Cubesat
2021-01-06 19:24:16
Equal!!!
Equal!!!
Mathfan9
2021-01-06 19:24:16
bigger
bigger
Zanderman
2021-01-06 19:24:16
equal?
equal?
Tong.Qiu
2021-01-06 19:24:16
equal?
equal?
goodskate
2021-01-06 19:24:16
=
=
Mathdreams
2021-01-06 19:24:16
Equal
Equal
BlueyStudiosYT
2021-01-06 19:24:16
bigger
bigger
bdcl
2021-01-06 19:24:16
same?
same?
Hornet
2021-01-06 19:24:16
still equal
still equal
BananaChamp1
2021-01-06 19:24:16
Bigger?
Bigger?
scube20
2021-01-06 19:24:16
equal
equal
Pinepiano
2021-01-06 19:24:16
equal
equal
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.
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.
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.
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.
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?
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.
The answer is EQUAL. Infinity times infinity still equals infinity.
miyomiyo
2021-01-06 19:27:05
Want to see a proof?
Want to see a proof?
NG2021
2021-01-06 19:27:18
yes
yes
rench23sugar
2021-01-06 19:27:18
yes!
yes!
puppyjohnson
2021-01-06 19:27:18
ya
ya
Mathfan9
2021-01-06 19:27:18
yes please
yes please
ploik11
2021-01-06 19:27:18
yes
yes
pinkpig
2021-01-06 19:27:18
YES!
YES!
miyomiyo
2021-01-06 19:27:22
This is called the "snake proof" and I think it's pretty neat.
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.
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.
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.")
("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?
So there you go, QED. Does everyone find this proof convincing?
jfelicita
2021-01-06 19:29:25
Ohhh..
Ohhh..
ianlee25510
2021-01-06 19:29:25
Cool times infinity!
Cool times infinity!
firebolt360
2021-01-06 19:29:25
Oh thats pretty neat!
Oh thats pretty neat!
edgymemelord
2021-01-06 19:29:25
yeah
yeah
JaiT
2021-01-06 19:29:25
yes
yes
STGpotter
2021-01-06 19:29:25
yup
yup
emma_maths
2021-01-06 19:29:25
Yes!
Yes!
amalgam1971
2021-01-06 19:29:25
yes
yes
brendan_cape
2021-01-06 19:29:25
Yeah
Yeah
miyomiyo
2021-01-06 19:29:38
Great, so it's settled. Infinity times infinity equals infinity.
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?
Are we ready to give up? What do you all think: Is there something bigger than infinity?
ianlee25510
2021-01-06 19:30:22
nope
nope
KT22
2021-01-06 19:30:22
no
no
yitesr
2021-01-06 19:30:22
no
no
neev12
2021-01-06 19:30:22
no!
no!
Joshua_Oh
2021-01-06 19:30:22
No there is nothing bigger than infinity
No there is nothing bigger than infinity
iqzhang
2021-01-06 19:30:22
maybe
maybe
pinkpig
2021-01-06 19:30:22
Nothing is bigger.
Nothing is bigger.
JaiT
2021-01-06 19:30:22
maybe..
maybe..
OpalRaven
2021-01-06 19:30:22
$Nope!$
$Nope!$
Cubesat
2021-01-06 19:30:22
Yes!!!
Yes!!!
Tong.Qiu
2021-01-06 19:30:22
I'm pretty sure there is something bigger
I'm pretty sure there is something bigger
NG2021
2021-01-06 19:30:22
nope, there is nothing bigger than infinity
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?
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
YES.
YES.
aops-g5-gethsemanea2
2021-01-06 19:30:54
ready!
ready!
pinkpig
2021-01-06 19:30:54
YES!
YES!
mathKidMom
2021-01-06 19:30:54
yes
yes
RRao56
2021-01-06 19:30:54
TELL US!
TELL US!
miyomiyo
2021-01-06 19:31:04
There IS something bigger than infinity!
There IS something bigger than infinity!
miyomiyo
2021-01-06 19:31:20
It's called "the continuum."
It's called "the continuum."
miyomiyo
2021-01-06 19:31:28
We usually write it with a fancy lowercase c, like this: $\mathfrak{c}$
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}.
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: $\mathfrak{c} > \infty$.
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?
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.
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.
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.
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 $\mathfrak{c}$?
Which of these pictures represents $\infty$ and which represents $\mathfrak{c}$?
sosiaops
2021-01-06 19:33:43
c is on top
c is on top
Ron.Livne
2021-01-06 19:33:43
the top one is c
the top one is c
turtle22
2021-01-06 19:33:43
bottom is infinity
bottom is infinity
STGpotter
2021-01-06 19:33:43
c represents the first one
c represents the first one
awesomeness07
2021-01-06 19:33:43
top is c and bottom is infinity
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!
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 $\mathfrak{c}$?
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
infinity!
infinity!
Blossom1
2021-01-06 19:34:29
infinty
infinty
discula2020
2021-01-06 19:34:29
infinity
infinity
goodskate
2021-01-06 19:34:29
infinity
infinity
MLiang2018
2021-01-06 19:34:29
infinity
infinity
ThinkingThings
2021-01-06 19:34:29
infinity
infinity
Tong.Qiu
2021-01-06 19:34:29
infinity
infinity
pyz18
2021-01-06 19:34:29
infinity
infinity
miyomiyo
2021-01-06 19:34:46
Yep, $\infty$. And how many positive integers are there total — positive, negative, and zero? $\infty$ or $\mathfrak{c}$?
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
still infinity
TheSubway
2021-01-06 19:35:21
Infinity.
Infinity.
NonOxyCrustacean
2021-01-06 19:35:21
infinity
infinity
EandCheckmark
2021-01-06 19:35:21
infinity
infinity
SparklyFlowers
2021-01-06 19:35:21
still infinity
still infinity
mathKidMom
2021-01-06 19:35:21
infinity
infinity
prickly-pear
2021-01-06 19:35:21
infinity?
infinity?
Tong.Qiu
2021-01-06 19:35:21
infinity still
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, $\infty + \infty = \infty$.
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?
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
c
c
TheSubway
2021-01-06 19:36:19
$\mathfrak{c}$
$\mathfrak{c}$
Tong.Qiu
2021-01-06 19:36:19
now it's c
now it's c
snow_monkey
2021-01-06 19:36:19
$\mathfrak{c}$
$\mathfrak{c}$
Quentissential
2021-01-06 19:36:19
$\mathfrak{c}$
$\mathfrak{c}$
UnknownMonkey
2021-01-06 19:36:19
$\mathfrak{c}$
$\mathfrak{c}$
duo_duo
2021-01-06 19:36:19
c
c
GabeW
2021-01-06 19:36:19
$\mathfrak{c}$
$\mathfrak{c}$
Cubesat
2021-01-06 19:36:19
$\mathfrak{c}$
$\mathfrak{c}$
HappyLemon07
2021-01-06 19:36:19
the fancy c
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.
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.
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.
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?
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
it has leftovers when compared to infinity
TheSubway
2021-01-06 19:38:26
That there are leftovers after pairing.
That there are leftovers after pairing.
ThinkingThings
2021-01-06 19:38:26
See if there's leftovers
See if there's leftovers
snow_monkey
2021-01-06 19:38:26
theres leftovers after pairing
theres leftovers after pairing
scube20
2021-01-06 19:38:26
making sure there is left over on the c side
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.
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.
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.
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?
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.
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
Yes!!
Yes!!
NG2021
2021-01-06 19:41:28
yes
yes
LightningStar
2021-01-06 19:41:28
yea
yea
Elektrolyte
2021-01-06 19:41:28
yes!
yes!
SparklyFlowers
2021-01-06 19:41:28
YES
YES
fangxcui
2021-01-06 19:41:28
yep
yep
masadca
2021-01-06 19:41:28
Yes!
Yes!
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!)
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.
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.
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.
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.
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
But, what if its in the middle
Cubesat
2021-01-06 19:43:08
What if it's in the middle?
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.
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.
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?
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
LL...
LL...
Cubesat
2021-01-06 19:45:20
$LLLLL\dots$
$LLLLL\dots$
dan09
2021-01-06 19:45:20
LL... to infinity
LL... to infinity
Mathdreams
2021-01-06 19:45:20
LL...
LL...
Ariva
2021-01-06 19:45:20
LLL...LLL...
LLL...LLL...
StickyWashington
2021-01-06 19:45:20
LL... repeating forever
LL... repeating forever
miyomiyo
2021-01-06 19:45:30
Yep: that point is named $LLLLLLLL\ldots$ (continuing forever).
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?
What about the point that's three-quarters of the way to the right?
yekolo
2021-01-06 19:46:22
RM
RM
URcurious2
2021-01-06 19:46:22
RM
RM
TheSubway
2021-01-06 19:46:22
$RM$
$RM$
ChiaravalleLearner
2021-01-06 19:46:22
RM
RM
edgymemelord
2021-01-06 19:46:22
RM
RM
JCJC
2021-01-06 19:46:22
RM
RM
Equinox8
2021-01-06 19:46:22
RM
RM
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.
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.
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.)
(This one's pretty tricky.)
boing123
2021-01-06 19:48:23
LRLRLRLRLR...
LRLRLRLRLR...
yekolo
2021-01-06 19:48:23
LRLRLR....
LRLRLR....
JazzyTheJazzer1234
2021-01-06 19:48:23
LRLRLR...
LRLRLR...
ARSM2019
2021-01-06 19:48:23
LRLRLRLR...
LRLRLRLR...
Cubesat
2021-01-06 19:48:23
$LRLRLRLRLRLRLRL\dots$
$LRLRLRLRLRLRLRL\dots$
Mathdreams
2021-01-06 19:48:23
LRLRLR...
LRLRLR...
speedstar
2021-01-06 19:48:23
LRLRLRLRLRLRLRLRLRLRLRLRLRLR...
LRLRLRLRLRLRLRLRLRLRLRLRLRLR...
goodskate
2021-01-06 19:48:23
LRLRLRLRLRLR...repeating forever
LRLRLRLRLRLR...repeating forever
NonOxyCrustacean
2021-01-06 19:48:23
LRLRLRLRL....
LRLRLRLRL....
miyomiyo
2021-01-06 19:48:50
That one would be named $LRLRLRLR\ldots$ (repeating forever).
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.
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.
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.
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.
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?
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?
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
make a new address
timchen
2021-01-06 19:52:31
find something that is missing
find something that is missing
URcurious2
2021-01-06 19:52:31
find a point not on the list
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
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.
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.
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?
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."
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!
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.
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.
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?
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?
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
No
No
ARSM2019
2021-01-06 19:58:16
no
no
Mathfan9
2021-01-06 19:58:16
no
no
mathKidMom
2021-01-06 19:58:16
no
no
JazzyTheJazzer1234
2021-01-06 19:58:16
no
no
prickly-pear
2021-01-06 19:58:16
can't be
can't be
Cubesat
2021-01-06 19:58:16
Nope! The first letter is different.
Nope! The first letter is different.
bapmookja
2021-01-06 19:58:16
no because the first letter is different
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.
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?
Is it the second item in the list?
pooja26678
2021-01-06 19:58:59
no
no
cutesamoyeds
2021-01-06 19:58:59
no
no
bapmookja
2021-01-06 19:58:59
no because the second letter is different
no because the second letter is different
Quentissential
2021-01-06 19:58:59
No, disagrees on the second letter
No, disagrees on the second letter
mathKidMom
2021-01-06 19:58:59
nope
nope
t_ameya
2021-01-06 19:58:59
no
no
goodskate
2021-01-06 19:58:59
no because second would be different
no because second would be different
krishgarg
2021-01-06 19:58:59
No, the second letter must be different
No, the second letter must be different
Equinox8
2021-01-06 19:58:59
no because they disagree on the 2nd letter
no because they disagree on the 2nd letter
miyomiyo
2021-01-06 19:59:10
Is it the $n$th item in the list?
Is it the $n$th item in the list?
pooja26678
2021-01-06 19:59:39
NO
NO
Mathfan9
2021-01-06 19:59:39
no, the nth letter would be different
no, the nth letter would be different
URcurious2
2021-01-06 19:59:39
No
No
mt2021
2021-01-06 19:59:39
no because they disagree in the nth letter
no because they disagree in the nth letter
Quentissential
2021-01-06 19:59:39
No, it disagrees on the nth letter
No, it disagrees on the nth letter
mathKidMom
2021-01-06 19:59:39
no
no
RRao56
2021-01-06 19:59:39
no they disagree on the nth letter
no they disagree on the nth letter
Cubesat
2021-01-06 19:59:39
Nope!
Nope!
krishgarg
2021-01-06 19:59:39
No, the nth letter must be different
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.
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.
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
haha take that rival
mathKidMom
2021-01-06 20:01:09
yay!
yay!
jfelicita
2021-01-06 20:01:09
Take that, imaginary rival
Take that, imaginary rival
RRao56
2021-01-06 20:01:23
What if they add it to their list
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?
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
Do it again
HappyLemon07
2021-01-06 20:02:02
then you do the same thing
then you do the same thing
mt2021
2021-01-06 20:02:02
we do the same thing again
we do the same thing again
ColtsFan10
2021-01-06 20:02:02
then we can get a new one
then we can get a new one
erm999
2021-01-06 20:02:02
do the same thing again
do the same thing again
EandCheckmark
2021-01-06 20:02:02
WANNA SEE ME DO IT AGAIN???
WANNA SEE ME DO IT AGAIN???
Lily55
2021-01-06 20:02:02
Just use the diagonal argument again.
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.
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.
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!
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?
And what do we say at the end of a proof?
goodskate
2021-01-06 20:04:11
QED
QED
pyz18
2021-01-06 20:04:11
QED
QED
blue_Yonder
2021-01-06 20:04:11
QED
QED
jfelicita
2021-01-06 20:04:11
QED!!!
QED!!!
Mathfan9
2021-01-06 20:04:11
Q.E.D
Q.E.D
pyz18
2021-01-06 20:04:11
QED, boom
QED, boom
Korra
2021-01-06 20:04:11
MIC DROP!
MIC DROP!
Quentissential
2021-01-06 20:04:11
Q.E.D.
Q.E.D.
URcurious2
2021-01-06 20:04:11
$QED$
$QED$
prickly-pear
2021-01-06 20:04:11
QED
QED
miyomiyo
2021-01-06 20:04:18
QED!
QED!
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.
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."
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.
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
$\aleph_0$ 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 $\mathfrak{c} > \aleph_0$.
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?
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?
is there anything bigger than continuum?
Ningster
2021-01-06 20:08:54
is there an infinity between the two.
is there an infinity between the two.
Elektrolyte
2021-01-06 20:08:54
What's the greatest form of infinity?
What's the greatest form of infinity?
yekolo
2021-01-06 20:08:54
How many infinities are there?
How many infinities are there?
ARSM2019
2021-01-06 20:08:54
are there even bigger infinities?
are there even bigger infinities?
prickly-pear
2021-01-06 20:08:54
is there a biggest infinity?
is there a biggest infinity?
neev12
2021-01-06 20:08:54
all the different sizes
all the different sizes
jfelicita
2021-01-06 20:08:54
How many different sizes of infinity are there?
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.
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!
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 $\mathfrak{c}$?
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.
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 $\mathfrak{c}$!
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?
How is this proved?
ab2024
2021-01-06 20:11:30
how do we know it's impossible to prove?
how do we know it's impossible to prove?
goodskate
2021-01-06 20:11:30
wait... what????
wait... what????
yekolo
2021-01-06 20:11:30
How is it proved that this is impossible to prove?
How is it proved that this is impossible to prove?
snow_monkey
2021-01-06 20:11:30
that's a proven fact?
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!!
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!
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.
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?)
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.
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: https://www.penguinrandomhouse.com/books/609844/math-without-numbers-by-milo-beckman/
You can order the book here: https://www.penguinrandomhouse.com/books/609844/math-without-numbers-by-milo-beckman/
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: https://www.youtube.com/channel/UCURP6q1VjGEFfy7GPP3Tm0w
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
miyomiyo
2021-01-06 20:15:42
That's all. Now that we're done, you can go ahead and type numbers again!
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!
Yay! I love math books!
ThinkingThings
2021-01-06 20:16:18
1+1=2
1+1=2
adatta0517
2021-01-06 20:16:18
34567890987654323456789
34567890987654323456789
kpatel0581
2021-01-06 20:16:18
12343
12343
bronzetruck2016
2021-01-06 20:16:18
42
42
mjjiang
2021-01-06 20:16:18
23
23
Equinox8
2021-01-06 20:16:18
34
34
URcurious2
2021-01-06 20:16:18
1
1
URcurious2
2021-01-06 20:16:18
12
12
mathKidMom
2021-01-06 20:16:18
1290830194037598308925029384
1290830194037598308925029384
vapodaca
2021-01-06 20:18:52
Thanks everyone for participating!
Thanks everyone for participating!
vapodaca
2021-01-06 20:19:14
And thank you Milo, for leading this fun discussion!
And thank you Milo, for leading this fun discussion!
Maxlovesairandteslas1
2021-01-06 20:21:16
Bye!
Bye!
Pikachu19
2021-01-06 20:21:16
THANK YOU
THANK YOU
Blossom1
2021-01-06 20:21:16
bye! thank you!
bye! thank you!
Epicsmartypants
2021-01-06 20:21:16
thx bye
thx bye
superm8
2021-01-06 20:21:16
Thank you!
Thank you!
hashtagmath
2021-01-06 20:21:16
Good night!
Good night!
Maxlovesairandteslas1
2021-01-06 20:21:16
byebye
byebye
zby
2021-01-06 20:21:16
bye bye!
bye bye!
Maxlovesairandteslas1
2021-01-06 20:21:16
Thank you!
Thank you!
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