[MUSIC PLAYING] There's more than one size of infinity.

In fact, there are infinitely many sizes of infinity.

Lots of things come in different sizes.

Five is bigger than four.

Humans are bigger than mice.

Infinity is no different.

Some infinities are bigger, and some are smaller.

It isn't just one monolithic concept.

It occurs in different forms and sizes.

So what are all these infinities?

The smallest kind of infinity is the natural numbers-- 0, 1, 2, 3, 4, and so on.

Some people call these the counting numbers.

Your intuition might say that the size of the even numbers is a smaller infinity than that of the natural numbers.

We got rid of all the odd numbers, which feels like it shrinks the set, but the even numbers and the natural numbers are actually the same size.

Intuition in mathematics is tricky and often misleading.

To avoid being led astray, mathematics uses precise definitions and rigorous logic.

We have to figure out exactly what it means for an infinity to be bigger than another infinity.

So how do we do this?

With finite quantities, it's simple.

If I gave you two bags of pennies and ask which one was bigger, you'd just count the pennies.

Easy.

But we can't count to infinity, not even the smallest infinity.

So we have to find a way of telling which infinity is bigger that doesn't rely on counting.

We're going to have to use something called a bijection.

What's that?

Here's an example.

Let's say you're sitting on a bus and notice that all the seats are taken, but no one's standing.

Then you know that the number of people on the bus is equal to the number of seats on the bus.

No counting needed.

This is what mathematicians call a bijection.

A bijection is a way of pairing elements from one set with elements from another set, and it proves that the two sets are the same size.

There's a bijection, or pairing, between the seats and people.

Each person is paired with the seat they're sitting in.

It's also how I know I have the same number of fingers on both hands.

I can pair them up.

Got it?

If you can pair up two sets, they're the same size.

Let's go back to infinities.

The natural numbers and the even numbers are the same size of infinity because there is a bijection between the two sets.

Each natural number is paired up with two times itself.

Everyone has a buddy.

We kicked out all the odd numbers, but the set didn't get smaller.

That's part of the weirdness of infinity.

Here's another counterintuitive fact, the natural numbers are also the same size as the integers.

The integers are all the whole numbers, including the negative ones.

Here's the bijection that proves they're the same size.

It helps not to think of bijections as a formula, but as a rule for pairing things in one set with things in another set.

This diagram shows you the pattern, or rule, for pairing every integer with a natural number.

The integers include all the natural numbers plus all the negative whole numbers, but we can pair them up exactly, so they must be the same size.

So the even numbers and the integers belong at the bottom of the tower of infinities with the natural numbers, and as I said way back in the beginning of this episode, natural numbers are the smallest kind of infinity.

But are there any bigger sizes of infinity?

What goes on the tower above the natural numbers?

The real numbers, that is all the numbers on a number line including those with infinitely many decimal places-- 2/3, pi, 1 million, all of them.

They belong further up the tower.

You can choose to believe me, or if you want a technical proof, click on the link in the description where I lay it out for you.

For now, let's stay focused on the task at hand understanding the hierarchy of infinities.

An interval on the real number line is also an infinite set.

Intuitively, we tend to think that the interval from 0 to 10 is twice as big as the interval from 0 to 5.

But amazingly, in terms of the tower of infinities, all intervals are the same size.

They're all as big as the real numbers.

Here's the bijection that shows that the interval from 0 to 1 is the same size as the real numbers.

You can use the same idea to show that any interval is the same size as the entire real number line.

We bent the interval into a semicircle.

Then each ray extending from the center of the semicircle pairs up a point on the interval with a point on the real number line.

Again, everyone has a buddy, so it's a bijection.

The interval 0, 1 is the same size as all the real numbers, and so they belong on the same spot on the hierarchy of infinities.

So now we have a smallest infinity, the natural numbers, and a bigger one, the real numbers.

The famous 19th century mathematician, Georg Cantor, wondered if there were any sizes of infinity that are between the natural numbers and the real numbers, and he noticed that it doesn't seem likely.

Like Cantor, we tested a bunch of sets that seem to be between the naturals and the reals, but the integers were smaller than expected, the same size as the natural numbers, and an interval was bigger than expected, the same size as the real numbers.

These observations lead Cantor to the continuum hypothesis-- there's no size of infinity between the natural numbers and the real numbers.

In other words, the natural numbers are the first block, and the real numbers are the second block in the tower of infinities, and there's nothing in between.

But here's the unexpected part about the continuum hypothesis.

Decades after Cantor's conjecture, mathematicians proved that the continuum hypothesis is independent of Zermelo-Fraenkel set theory with choice.

In English, this means that the continuum hypothesis cannot be proved or disproved using the standard rules of mathematics, which is kind of crazy.

Math is really in to rules.

That's kind of what some people don't like about it, but through creative applications of these rules, mathematicians have built an incredible and complex structure of ideas.

The most basic set of nine rules, ZFC, can be used to prove statements like 2 plus 3 equals 5, and disprove statements like 2 plus 3 equals 6.

These rules determine so many facts about our mathematical universe, but these rules cannot prove the continuum hypothesis, which would mean showing that the real numbers are the next size of infinity after the natural numbers, and they cannot disprove the continuum hypothesis, which would mean showing that there are sizes of infinity between the natural numbers and the real numbers.

In one model of the tower of infinities, the real numbers sit directly above the natural numbers, like this.

But in other models, there are many infinities in between, like this or this.

The rules of math don't say that one tower is correct and the others are wrong.

Mathematics proves so many things, simple statements like 2 times 4 is 8, and complex ideas like how to optimally pack spheres in eight-dimensional space.

But despite all it's capable of proving, mathematics, as it stands, is surprisingly agnostic with regards to which hierarchy of infinities is correct.

But if you have an opinion about it, let us know in the comments.

We'll see you next week on Infinite Series.

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