How to Count Infinity

"Some infinities are bigger than other infinities" - Hazel Grace Lancaster, in "The Fault in Our Stars," by John Green

And twitter - @minutephysics

Minute Physics provides an energetic and entertaining view of old and new problems in physics -- all in a minute!

Music by Nathaniel Schroeder

Thanks to Perimeter Institute for support.

[starcubey]

0:25
"counting sheep is boring"
me: tru
"that is, unless you want to count infinitely many sheep"
me: ...isn't that even more boring???

[Voltorb]

I... give me a minute, my brain is reloading.

[MuffinsAPlenty]

Actually, numberphile's explanation does describe it as a "bigger" type of infinity, too.
And there's a good reason for that. In mathematics, we often like to put some type of order on sets. For example, the natural numbers are ordered and we determine this order, more or less, based on addition.
Cardinalities ("sizes" of sets) are ordered based on injections. If there is an injection from set A to set B, then |A| ≤ |B|.
(continued)

[monkeydog]

I just just finished my 6 hr review in calculus.. After watching this, i can't think anymore. Brain is overloaded.

[CharlesPanigeo]

In technical language, we say that two sets have the same cardinality if there exists a bijection between them, that is, two sets A and B have the same cardinality if there exists a function from A to B that is both one-one and onto. One-one means that every element in the set A maps to a unique element in set B( no two elements in A map to the same element in B). Onto means that every element in B is mapped to by some element in A.

We assign the cardinality of the natural numbers, { 1, 2, 3, 4, ...} as Aleph-0 (aleph naught) and we say that a set is denumerable if it has the same cardinality as the natural numbers.

We can prove using Cantor's Diagonal Argument that there is no bijection from the natural numbers to the closed interval (0, 1), which Henry does in this video, so they do not have the same cardinality. However, since there does exist a one-one function from the natural numbers to (0, 1), the cardinality of (0, 1) must be greater than or equal to that of the natural numbers. Since it has to be greater than or equal to, but we proved it's not equal to, the cardinality of (0, 1) must be strictly greater than the cardinality of the natural numbers.
So, the open interval (0, 1) is not denumerable.

We can also prove that there exists a bijection from (0, 1) to any open interval on the real number line (a, b) where a<b. So, there exists a bijection from (0, 1) to (-π/2, π/2). There also exists a bijection from (-π/2, π/2) to all of the real numbers, which is f(x) = tan(x). Since the composition of bijections is also a bijection, there exists a bijection from (0, 1) to the real numbers. So (0, 1) and the real numbers have the same cardinality. However, since the cardinality of (0, 1) is greater than the cardinality of the natural numbers, the cardinality of the real numbers is greater than the natural numbers. This is what people mean when they say "there are more real numbers than natural numbers."

-Charles Panigeo, mathematics major at the Univeristy of Alaska

[MadNotAngry]

Damn, I can't use words to describe what's going on in my head. As one of your lessons starts to penetrate my skull, you introduce another point that just expands my understanding further. Watching another video makes me think about other lessons, and soon enough, my mind's blown...
.
Thanks for that.

[danerdYT]

The comments section in this video:

33% - People who have no idea what is going on and are willing to admit that
33% - People who have no idea what is going on, but pretend that they are smarter than everyone else and try to disprove the video
33% - TFIOS
1% - People who actually understand what's going on

[trojantm]

you are saying that the infinity between 0 and 1 is greater than the infinity of all real numbers. which may be true! idk, but the way you explained it makes literally no sense. you claim that since you can always find a new number between 0-1 for any new integer that that there must be more between 0 and 1 than integers, but can you not go the other way around? for any new 0.whatever you fine, you can just count more more integer for that new decimal. your example works in both directions and can't be used to say that there is more infinity between 0-1 than the infinity of all integers.

[danielnavarro]

Maybe it's a little less formal, but I think it's easier to see there are more real numbers between 0 and 1, than there are integers using fractions.
1/1, 1/2, 1/3, ..., 1/n, ...
There you can have all the integers in the denominator without covering all the numbers between 0 and 1, like 2/13.

[veronicachow]

the bass in the background music sounds awesome

and i finally understand the meaning of infinity now thx a lot

[renardleblanc]

"When we say we have 5 sheep, what we really mean is we have 1 sheep for every number from 1 to 5." The most eloquent and concise explanation of counting that I have ever heard. X3

[whyamihere]

Wow, this actually gave me a new perspective on infinity... :D

[TheIzzy]

The Fault in our stars is a fantastic book. John Green is incredible. Great video!

[ultimateredstone]

One infinity seems to offer more "labels", made by us to do maths, but does that really mean one infinity is bigger than another? It just has more "types" of information, but the amount of information is the same in both infinites.

[iustinniculae]

So if I want to count infinity, should I simply count to 1...?

[Naitasm]

For everyone getting confused, it's not exactly the case that one infinity is BIGGER than the other, it's that one (the infinite set of natural numbers) is countable, while the other (infinite set of numbers between 0 and 1) is uncountable. This is the very definition of countable - each number can be related to either the subset or the full set of natural numbers. That's similar to saying uncountable sets are "bigger than" countables, but infinities are weird, I don't think "bigger than" applies to them. It's Cantor's Diagonal Argument, if anyone wants to look into it more.

We learnt that in Uni, but there's one thing that I've never found an answer to, I was wondering if someone could give me one. If you did this and mapped all natural numbers to some real numbers, then constructed a new real number that was different to every other one (as he does, just between 0 and 1), couldn't you then just remove the decimal point on every single real number and have a set of natural numbers on the right-hand side? Can't you just use this exact same proof, therefore, to show that there are "more" natural numbers than there are natural numbers? Obviously not, since I think that would mean the theory is wrong, but I have no idea why I'm wrong to say that we CAN do that.

[lostchimaera]

Absolutely love the The Fault In Our Stars reference

[fitforlife]

I bet most people that comment on these videos and call them wrong have never taken any math or physics classes past high school lol. 

[kenjkun]

For some people that may be confused, even tho there are more real numbers between 0 and 1 than the integers, but there are still as many numbers in the naturals as in the rationals

[ragnkja]

Why haven't I re-watched this since reading TFiOS? That reference, which practically jumped at me now, was completely lost on me the first time! (Also, the way you address her as "Hazel Grace" is adorable.)