Hello there! Lovely to meet you. I've been reading a lot about infinity lately and
guys it is so cool. We all kind of have a concept of infinity, like something that
goes on forever and ever and ever. If you try to count to the biggest number you
can think of you can always add one more. The idea itself isn't that hard to grasp,
but when you try to make it rigorous like mathematics likes to do that's
where things get tricky. It turns out infinity comes in different sizes. This
german guy named Georg Cantor found that there are different kinds of infinity
and some are bigger than others. To understand this we actually need a lot
of background information but the background information is really cool
too. We'll start with Set Theory. A set is basically a group of things. This is a
set of four dogs. This is a set of four forks. The number of things or elements
in a set is called its cardinality, so these sets both have a cardinality of
four. Now when two sets have the same cardinality they're said to be the same
size. The size is defined in a bit of a strange way. The normal intuitive way to
think about it would be to say that both of these sets have four elements,
therefore they're the same size. This definition works well for finite sets
but when we start dealing with infinite sets it isn't good enough.
So same size is defined as the ability for the elements of both sets to be
paired up with each other. You'll see why this is important... right now! Let's look
at a set with infinite cardinality: the set of natural numbers. It starts at one
or sometimes zero depending on what text book you're reading, and increases by one
forever and ever and ever. This is totally allowed in set theory by
the way, we're not breaking any rules. Now let's look at another infinite set: the
set of squares of the natural numbers. Every element in the second set, the set
of squares, can be paired to a number in the first set, the set of natural numbers.
So then according to our definition they're the same size. But intuitively
this feels wrong. There are so many more numbers in the set of natural numbers
than the set of squares. Is one set bigger than the other? But they're both
infinite... infinity is infinity, right? It was at this point that Georg Cantor
came along to save the day. He thought a lot about infinities and actually legend
has it that he thought about them so much that he drove himself insane,
spending his last 20 years in and out of mental institutes. But here, we'll remember
him for his amazing work on numbers. He introduced the idea of an infinitely
enumerable set. This just means a set whose elements can be paired off with
the natural numbers. The word typically used is "counted" because effectively
that's what pairing things off with the natural numbers is, but that's kind of
confusing because you can't really count all the natural numbers. You'd die around
three billion. So some prefer to call it "listing" and I rather like this. Cantor
declared that infinitely enumerable sets are all the same size. The set of
natural numbers is obviously enumerable, as it can be paired off with the set of
natural numbers. The set of squares is enumerable, as we showed before. So then
it's the same size as the set of natural numbers. But are there some infinite sets
that can't be listed? Are there different sizes of infinity? Cantor suspected
there were. In particular he wondered about the relationship between the set
of natural numbers and the set of real numbers. Okay we need to take a break
from set theory to talk about number theory, but don't worry, it's just as
fascinating. To understand Cantor's work we first need to understand the
different types of numbers and their relationship to each other. We've already
talked about the natural numbers, whole numbers from 1 or 0 to infinity. Then
there are the integers. These are the natural numbers with the addition of the
negatives. They increase all the way to infinity and decrease all the way to negative
infinity. Then there are the rational numbers. These are just numbers that can
be represented by a ratio of two integers. You might know them as
fractions. All rational numbers can also be expressed as a decimal with their
either being a finite number of decimal places or a repeating pattern. Then there
are the irrational numbers, which can't be expressed as a ratio of two numbers.
Root two is a common example and here's the proof of its irrationality if you're
interested. I think it's pretty clever. If you express an irrational number as a
decimal the digits keep going on forever and ever with no apparent pattern.
They're also sometimes called surds which is actually from the word absurd
because the ancient Greeks really didn't like them.
According to legend Pythagoras' disciple Hippasus was the one who
figured out that root 2 is irrational and to celebrate his finding the ancient
Greeks threw him into the Mediterranean Sea, like, to drown. Yeah they didn't like
irrationals. Then there are the algebraic numbers. These are just the
numbers which are solutions to algebraic equations, you know, these things. They
have one variable, usually x, and a series of terms that sum to 0, in this case, three.
Negative 3 is a solution to this equation
so it's an algebraic number. All integers and rational numbers are algebraic
numbers, a lot of irrationals are too. So are there numbers that aren't solutions to
any equation like this? Well yes there are. In fact two of the most famous
numbers, pi and Euler's number e can't be calculated by any equation. These types
of numbers have the coolest name of all: transcendental numbers, called so because
they transcend the algebraic. All of the numbers I've talked about make up the
real numbers. Here's a diagram to clarify things. There are also the imaginary
numbers but they're not important here. Okay then now that we know about the
reals and all of the numbers they're made of we can get back to infinity. Now
Cantor wanted to know whether the set of real numbers was enumerable. That was,
could every real number be paired off with a natural number just like the set
of squares. Well let's take small steps shall, we but like not in a patronizing
way... Can the set of integers be paired off with the natural numbers? This isn't the
usual way one would write out the integers but if we keep going with this
pattern we'll eventually write them all out, so yes the set of integers is
infinitely enumerable. How about the set of rational numbers? Surprisingly they
too are enumerable. If we write them out in this rather unconventional way we can
see that with the first element, the numerator and denominator add up to 2.
The next two elements have numerators and denominators that add up
to 3. The next three elements numerators and denominators add up to
4. A list that continues like this contains all the positive rational
numbers. We can just include the negative rational numbers too if we alternate like
we did with the integers. So then the set of rational numbers are enumerable.
Cantor even proved that all the algebraic numbers are enumerable. So
then what about the Reals, which encapsulate both the algebraic and the
transcendental? In the same paper he proved that the real numbers are not
enumerable, they are in fact a different kind of infinity, imaginatively called
non-enumerable. Cantor was a huge fan of proof by contradiction and he had an
inkling that the Reals were non- enumerable, so of course, he started the
proof with the assumption that they are. Imagine that each of the real numbers is
represented by an omega term, this funny looking w. So if we assume that
they're enumerable we can pair them off with each of the natural numbers. Cantor
showed that this list will always be incomplete. Pick any number, alpha and any
larger number, beta, and mark them out on the real number line. This is just a list
of real numbers marked out on a line. Now find the first real number on your list
that is bigger than alpha and the first real number that's smaller than beta,
we'll call them alpha prime and beta prime. Do this a couple more times. Now
the thing is you'll always be able to find two more numbers between the last
two numbers. Suppose you get stuck, where v indicates around a trillion prime
marks, but a finite number. Now no matter how hard you search through the list of
enumerated real numbers you can't find another one larger than alpha v and
smaller than beta v, so then are there no more reals between them? Well what about
alpha v + beta V on 2, the average? Your list is missing that number. And then
what about all the numbers between alpha v and the average and beta v and the
average? This could go on forever, it doesn't ever stop. There are an
infinity of numbers between alpha and beta and between alpha v and beta v. This
is called the infinity of the continuum and we can't pair it off with the
natural numbers the way we could with the integers and the rationals. There are
two types of infinities, the enumerable infinity of the natural numbers and the
non enumerable infinity of the continuum. I really like this proof because I think
it conveys the intuition really well but it is also pretty abstract, so there's
another proof I want to show you too, you know, just for fun. It's called proof by
diagonalization, or the diagonalization proof, or diagonalizing, it involves
diagonals. Cantor again begins by assuming that you can number all the
real numbers on a list. In fact he restricts it to just real numbers between
0 and 1. We can't really do this in order because the first number would be
a point followed by an infinite amount of 0s with a 1 at the end, which
would take an infinite amount of paper and ink to write, so let's just choose
randomly. These numbers are pretty good. It's got a quarter, a third, pi on 10, e on 10.
It's infinite but is it complete? Nope, and this is how Cantor proved it. He went
down the list in a diagonal fashion selecting the appropriate digit, for
example the first digit of the first number, the second digit of the second
number, the third digit of the third number and so on, and then he used them
to create a new number. Then he added 1 to every digit. Now is this new number on
the original list? Well let's do a rigorous check. Is it the first number? No
because the first digit of the first number is 1 and the first digit of the
second number is 2. All right then is it the second number? Well no because the
second digit of the second number is 5 and the second digit of the new number
is 6. It's not any of the numbers. We've found a number that isn't on the
original list, so let's add it on. Now is the list complete? Well you can just
repeat the process and find another number that wasn't on the list.
By coming up with new digits in this diagonal fashion we can always find
another number that isn't on the original list, therefore our list can
never be complete. There are always numbers that aren't on it. It's
impossible to list all the real numbers between 0 & 1, let alone 0 and infinity.
Cantor also showed that not only are infinities different, some are
infinitely bigger than others. The cardinality of an infinitely
enumerable set is smaller than the cardinality of the continuum, but like, a
lot smaller. But the only difference between the algebraic numbers which are
enumerable and the Reals which aren't are the transcendental numbers. So what
does that mean? It means that transcendental numbers make up pretty
much all possible numbers! Now I want to wrap up with a quote from the book I got
most of my information from. "As humans we value neatness order and patterns. We're
interested only in numbers that have meaning to us. From counting farm animals
we have invented the natural numbers, from measurement we have invented the
rational numbers, and from higher mathematics we have invented the
algebraic numbers. We have plucked all these numbers from the continuum while
ignoring the vast depth in which they swim like microscopic bacteria in the
ocean. Virtually every number is transcendental". I don't know I think
that's pretty cool. I must say this was probably my favorite video to make because
I found the topic so fascinating, but it was also rather difficult. There were a
lot of subtleties which took a while to sink in. Reading through theory is one
thing but if you really want to ground your understanding nothing beats working
through practical problems like the ones on brilliant org. That's what I did while
I was researching this video. Brilliant is a learning website which focuses on
math, physics and computer science. There's an awesome course on number
theory which extremely conveniently had a whole section on infinity. It has a lot
of concepts that were included in this video
like cardinality and the different kinds of infinity, and a lot that weren't like
showing infinite series through geometry, a way I'd never seen it taught before. It
was actually so cool to see a visual representation like that. A subscription
will get you access to all of these courses. Signing up is free and brilliant
is offering a 20% discount to their premium program to the first 200 people
who sign up using this link. Just go to brilliant.org/upandatom, link is
on screen and in the description. Thanks so much for watching guys! I know this
video was pretty long but I physically could not not include stuff it was all so
fascinating. It was actually a requested topic by one of my patrons Robert, hi
Robert! I hope you enjoyed watching the video as much as I enjoyed creating it. A
big thank you to all my patrons for your ongoing support and until next time, bye!