The REAL reason ∞² = ∞ | Set Theory Intro! #numbers #set #maths

Added: 2026-05-16

[00:00:00]Is infinity squared bigger than infinity? The answer is no, and I'm going to show you why. Infinity is represented by the size of the set. This is all the natural numbers, 1 2 3 etc. The number of them is what we call infinity. Now, what is infinity squared going to be represented by? We're going to look at n squared. Okay, so n squared is going to equal to n cross n, which is now going to be all the tuples, the set of all tuples of the form m comma n, where m and n are natural numbers, 1 2 3 etc. Which you can describe graphically in the plane as follows, by a lattice.

[00:00:39]So, you start off with 1 comma 1, then you have 2 comma 1, then you have 1 comma 2, etc. You have these dots, they extend in all directions forever, and you can sort of see there are infinite choices for the first coordinate, infinite choices for the second coordinate, so the size of this should represent infinity squared. But, as I said, infinity squared is really just infinity. How to prove that? Well, what I'll do is I'll show you can naturally match up the elements of n, the natural numbers, with the elements of n squared, which are these pairs, in a unique way, so that the correspondence matches a unique natural number with a unique element of n squared. How do we do that?

[00:01:17]We count n squared, so we start off with 1 1, that's the first element of n squared, then we go to 1 2, then we go to 2 1, then we go to this is going to be 3 1, then we go to 2 2, this is going to be 1 3, then we're going to go to 1 4, and so on and so forth. And you see that by doing this, we achieve what we want, we naturally match up n and n squared. You can think of this as a function f from n to n squared, and you can literally write down values. f of 1 is 1 1, f of 2 is going to be with the way we match it up is going to be 1 2, and f of 3 is going to be 2 1, and so on and so forth. So, infinity squared is actually infinity. I'm going to show you this stunning proof using prime numbers of the exact same fact. Click down below.

[00:02:03]You're going to love it. It uses prime numbers and properties of prime numbers to explain this phenomena, and it's also a beautiful introduction to more of the set theory, what it means to be countable, which in this case we say n squared is countable because you can naturally match it up with n. Click down below, smash that like button on this short, and have an amazing day. And I'm super excited to see you on my channel for tons more fun math content when you click that subscribe button as well.