What is Cantor’s diagonal argument?

Cantor’s diagonal argument shows that infinite sets can have different sizes, some infinities are bigger than others.

Imagine you're in a toy store with a huge shelf of colorful blocks. Each block has a unique number on it, like 1, 2, 3, and so on. Now, suppose someone claims they’ve matched every block to a different toy, one-to-one, and there are no leftovers. That would mean the number of blocks is equal to the number of toys.

But what if we had infinite blocks? You might think that means you could match them all up with infinite toys, but Cantor showed that’s not always true!

Let’s say someone tries to list out every possible block number in a never-ending list. It would look like this:

1. 0.123456...
2. 0.987654...
3. 0.111111...
4. 0.222222...

...

Now, here’s the diagonal trick: we go down the list, changing one digit from each number, like a diagonal line in a grid of numbers.

By doing this, we create a new number that wasn’t on the original list! That means there are more infinite numbers than we thought, some infinities are just bigger than others.