Infinity in mathematics: a disgrace to some mathematicians, and a greatly interesting concept to others. The well-respected German mathematician Johann Carl Friedrich Gauss (1777–1855) stated that infinity should never be permitted in mathematics. René Descartes (1595–1650) thought that because humans are finite beings, they should never have been able to come up with the concept of infinity. Instead, he proposed, it must have come to us through God, an infinite being.

Whatever your stance on infinity and its place in mathematics, it’s a marvelous concept that gets our heads spinning in circles. Humans weren’t meant to understand infinity. But that’s exactly why it’s so interesting!

So without furder ado, here are a few facts about infinity we love!

#### Counting small infinities

It might sound unintuitive, but there is such a thing as a countable infinity.

Let’s say you had a bag containing an infinite amount of chocolates. You can start counting these chocolates, because you can label them. You can label the first piece as piece 1, the second piece as piece 2, and so on. Of course, you won’t reach the end, because there is no end to this bag of delicacies. This might be what makes it sound unintuitive. But it doesn’t matter whether or not you reach the end, what mathematically matters is that you can put the chocolates in a one-to-one correspondance with the natural numbers (1,2, 3 …). This is what we call a countable infinity.

Here’s a question: are infinities always the same size? Your first reaction is probably yes. Whether you have a bag with infinitely many chocolates or infinitely many shoes, the amount in the bag is the same. But guess what? There are bigger things than an infinite amount of chocolate, and I’m not talking about infinite elephants!

#### Even bigger: uncountable infinities

There are infinities so large, we can’t count them, even if we counted forever. Let’s say you have a set of all real numbers, with a real number being a value that represents a quantity along a line. This line is called the real number line:

It’s impossible to count all the real numbers on this line: it’s uncountable infinity. Cantor’s diagonal argument, published in 1891, famously proved that when we construct a sequence, we can always construct a new real number that is not a part of the sequence yet. We do this by making sure that this new number fails to match every number on the list in (at least) one decimal place.

This is a very complex argument summed up in a few sentences (more information here , but it proves there is no countable list of real numbers which includes every real number and that the set of all real numbers is uncountable.

And then there are the irrational numbers. These are real numbers that cannot be expressed as a ratio of integers. Take for example: a 3 with an infinite amount of decimal numbers. It’s impossible to know what number comes after , because you would need an infinite amount of zeroes, followed by a 1. But you would never get to the 1, because of the infinite amount of zeroes. Which leads us to the following conclusion: we can never name the number preceding or following irrational numbers.

Irrational numbers cannot be counted because there is no successor function! This is why the set of all irrational numbers is a larger infinity than the set of rational numbers. And uncountable infinities are bigger than countable infinities. In other words: there are different sizes of infinity!

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#### Playing With Different Sizes of Infinity

You can make your brain do some real nice acrobatics if we play around with this even more.

Did you know that when you add an infinite number of elements to an infinite set, you will end up having an infinite set of same size? For example, the set of all integers is the same size as the set of even numbers (or odd numbers). In fact, if you have an infinite set of elements and you deduct an infinite number of elements, you will have an infinite set which has the same size as before.

Here’s another one: almost all numbers are not a power of five, but there are just as many numbers which are a power of five as there are natural numbers.

This is why the following funny example about monkeys and bananas is true.

**The Infinite Hotel Paradox**

Now that you understand all of this, you’ll have no problem grasping this fun video on the Infinite Hotel, a thought experiment created by German mathematician David Hilbert. It’s a lot of fun, and incredibly well animated and explained, so we recommend you watch it!

I think this is enough mindblowing for one post.

Do you have any other interesting, mindboggling infinity facts? Please let us know about them!

Let’s talk again soon,

Jeroen

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