Hey guys, how's it going? I uploaded a video last week called "my infinity is
bigger than your infinity" all about the different sizes of infinity and Cantor
sets. In it I went through two proofs about why the real numbers are not
infinitely countable or enumerable and there was a lot of confusion about the
first one. So the reason for the confusion was my fault. I tried to sketch
out the proof but I also didn't want the video to be too long so I tried to go
through it quite quickly but in doing that I left out some major details and
basically gave you an incomplete proof. I know there's nothing more frustrating
than feeling like you didn't understand something, especially when it's not your
fault, so I decided to make this video going through the proof again from start
to finish and hopefully that will clear any of the confusion and questions that
you have. So here we go. Suppose someone says they have a list of all the real
numbers. You want to step on their dreams and show them that their list is missing
a real number. Before we get into the details of how we're going to do this, to
avoid confusion I want to give the overall strategy of what we're doing.
Starting from an arbitrary interval on the real line the goal will be to shrink
that interval by using the real list in such a way that we can show that if the
process of shrinking doesn't converge to a number then all the reals can't be on
the list. Then we'll show that if it does converge to a number then either it must
take an infinite amount of time to do so, meaning that the number the interval
converges to can't be on the list, or it takes a finite amount of time to
converge, meaning that the list must be missing some other real numbers. If that
made absolutely no sense, don't worry, it will, I hope. All right let's get started.
Here we're defining a process for creating and shrinking an interval on
the real line. Start with two real numbers from the list, alpha 1 and a
larger number beta 1. Now find the first two numbers from the list that are
between alpha 1 and beta 1, alpha 2 and beta 2. Keep going with this pattern. For
any alpha and beta n find the first two distinct numbers from the list that are
between them, alpha and beta n plus 1. If you don't find a number the process
stops. If you only find one number between alpha n and beta
n, alpha n plus one is equal to beta n plus one. Now we have two series, the
alpha series and the beta series. We can take the limit of those series, in other
words what the alpha and the beta series converge to, and call them alpha infinity
and beta infinity. In a full mathematical proof we would have to show that the
series do in fact converge, but just take my word for it. Now these limits have to
be the same number. If there were a gap between alpha infinity and beta infinity
then there would be reals in between, for example alpha infinity plus beta
infinity divided by 2. So then the interval must converge to a point, which
means alpha infinity is equal to beta infinity. We'll call this number
eta, the funny-looking n. Now we're going to show two scenarios, one where the
series' converge to eta in a finite number of steps, and one where it takes
an infinite number of steps. And then show that both scenarios mean there are
some reals missing from the list. Suppose this real number eta is on the list,
which means it takes a finite number of steps to converge to it. We'll label this
finite number of steps m. Now look at the alpha and beta terms just before eta,
alpha m-1 and beta m-1. According to how the reals are defined
there should be an infinite amount of real numbers between them, like any
weighted average between the two. So then if we do find the real number eta,
that means we're missing all the real numbers between alpha m-1 and beta
m-1 except eta, so in the case where the series' converge to eta in a
finite number of steps and eta is on the list, there are real numbers
that aren't. So then what if it takes an infinite amount of steps to converge to
eta? Well that means there are an infinite amount of alphas and betas that
the process picked before it. Because of how we defined our process they must
have come before it on the list. To put it another way, say you asked your friend
"What natural number does eta correspond to on the list?", what would he
say? Infinity isn't an
answer and if he gives any natural number like 300 billion and 49, you can
say, "Well no it's not because there are an infinite number of alphas and betas
before it". Contrasting this to any enumerable set like the rationals, for
any number, your friend would be able to tell you the natural number that it
corresponds to. In the previous video we showed how every element in an
infinitely enumerable list could be mapped to the set of natural numbers. For
the set of reals we've shown this can't hold, so the real number eta cannot be
on the list, therefore the list is never complete and
the real numbers are non enumerable. So just to recap what we've done. There are
three critical cases we analyzed to show that the argument works. Case 1: the alpha
series and the beta series don't converge to the same limit, but this
means we're missing any real number in between the alpha and beta limit, so in
this case the list is missing reals. Case 2: the series' converge to the same limit
eta in a finite number of steps m, but this means that the alpha and beta
before eta alpha m-1 and beta m-1, which must be distinct, have an
infinite number of reals in between them which can't be on the list, otherwise the
process wouldn't have converged to the limit eta yet. So in this case there
are also reals that aren't on the list. Case 3: it takes an infinite amount of
steps to converge to the limit eta, but then there needs to be an infinite
amount of alphas and betas on the list before it, so eta itself cannot be on the
list. But why doesn't this argument work for enumerable steps like the rationals?
Well we're generating this number eta in a pretty complicated way. We're taking
the limit of two infinite series' and saying this number is what they converge
to, but the way the real numbers are defined means we could prove that eta is
real. But for the argument to work with rationals we would have to show that eta
is a rational number, but it turns out to be impossible to show that eta can be
expressed as a ratio of two integers. We need a more expressive number system to
include this abstractly defined number. What makes this argument so powerful is
that it's a self referential argument, which basically
uses the list to generate a number that can't be on it. So yeah that's the
complete proof. If you still have questions I've linked a bunch of sources
in the description. Again sorry for the initial confusion and thank you for your
patience. See you in the next video. Bye!