Are there more positive even numbers or positive numbers?
And by numbers, I’m talking about whole numbers, like 1, 2, 3, 721, and 600,053.
At first, this might seem like a trivial problem. Of course there are more positive numbers!
The set of positive numbers basically contains all the even numbers,
and some more! So it must be bigger.
But how do you find how big an infinite set is? For finite sets, it’s actually easy.
Look, which of these sets contains the most dogs.
Well, this one has two, and this one has seven, so the one on the right has more!
But if you try to do the same for the infinite sets of positive and even numbers, you will find
that you’ll be here for a very long time. There are an infinite number of positive numbers and an
infinite number of even numbers, so we’ll all be long gone before you finish counting all of them.
But if you’re satisfied with that explanation, the world where “of course there are less even
numbers than even and odd numbers combined”, you are free to take the blue pill and click away.
Never look back. No need to think about this world of infinities or sets or any of this nonsense.
Okay so here’s a thing.
Suppose I have a set containing three random numbers, let’s say… 3, 7, and 28. Three numbers
in the set, nothing special here. Now, let’s take every element in the set, and just add
1 to it. So instead of 3,7 and 28, we now have 4, 8, and 29. Cool, still three elements, right?
I mean, makes sense… Changing the contents of the set isn’t what's important, what matters
is the size of the set. The cardinality of the set, not the elements within it.
Okay, so that’s the setup. Here’s a fun trick. Remember this set of positive numbers? 1, 2, 3,
4, and so on. Right, okay… Take every element in that set, and double it. What do you get?
Wait... Uh... So we initially said that there are more positive numbers than there are even positive
numbers... but then we realized, that without ever changing the number of elements in the set,
we can simply just transform the set of positive numbers into the even numbers
just by doubling the individual elements of that set. Again, not changing the size of the set.
Now that’s problematic. A set cannot be smaller than itself, that doesn’t make sense.
Okay, but like, this is fun: Turns out like, adding numbers to these infinite sets don’t
change things either, like, right okay if I were to put in three extra numbers to this
list of positive numbers like let’s say, -2, -1, and 0, we might think okay, we now
have three extra elements so of course this set must be bigger than the one from before, right?
Umm… But then you can just add 3 to every element in the set,
and you basically just end up with the same set as you had before.
Uhh… Yeah… Infinite sets are tricky.
I mean, philosophically, what does it even mean to know the size of something that is infinite?
This is the kind of thing that mathematicians like to think about.
And then they come up with some fundamental rules that are used to then
spawn many many other branches of mathematics. And every so often someone realizes that all of
math is just wrong, and then we start over from scratch, that’s always fun.
Umm… That's a side line though…
Anyways, the current definition of cardinality, which a lot of originates with this guy,
deals with this concept called bijection, which is basically a fancy way of saying if
you can draw a mapping between two sets without repeats or overlap, then they’re the same size.
Great thing about this is that it works both with sets of finite and infinite size,
it’s kinda great!
That’s basically what we’ve been doing throughout this video. Just think of it like: Hey! I can just
take the element from the set on the left and map it to two times that number which appears on the
right, and if I can do that for every element without overlap, then they’re the same size.
And so, by the definition developed by Cantor,
the cardinalities of the set of even numbers and the set of natural numbers are the same.
Side Line: if you’ve ever heard of the “some infinities are bigger than others” line,
it derives from these ideas, same guy as before. I mean, there are some infinities that are so
large, that like you physically cannot, with infinite time and energy
find a mapping from this puny natural number infinity to this big bold
much larger infinity. I’m sure we’ll talk about that more in the future.
Anyways, if you want to learn more that’s great. Fortunately for you, there’s so much of this, this
isn’t even the tip of the iceberg. This is like, the snowflake that sits on top of the iceberg.
WELCOME TO MATHEMATICS. I AM YOUR GUIDE THROUGH THESE VAST NEW REALITIES.
If you want to leave now, that’s fine, unlike infinity life is short, I get it.
Otherwise, click subscribe and we’ll see how deep the rabbit hole goes. DO IT.