Cantor's diagonal argument

Imagine you have a lot of numbers, like 1, 2, 3, 4, and so on. Now, what if you have even more numbers? You might think you could never count all of them because there are just too many.

Cantor's diagonal argument is a way to show that some sets of numbers are bigger than others, even when they seem very large. It's like saying there are more stars in the sky than there are trees in a forest, even though both things seem really big.

To use Cantor's diagonal argument, we make a list of numbers, like this:

1. 0.123456789...
2. 0.246801357...
3. 0.987654321...
4. 0.112233445...
5. 0.314159265...
6. ...

Now, we want to prove that this list is missing some numbers, even though it seems to go on forever. To do this, we take the first digit of the first number (which is 0), the second digit of the second number (which is 3), the third digit of the third number (which is 7), and so on. We write down these digits in order:

0 3 7 4 5 9 ...

Now, we make a new number by changing each digit. If the digit is 0, we change it to 1. If it's anything else, we change it to 0. So we get:

1 0 2 5 4 0 ...

This number is different from every number on our list! That's because we made sure it's different in every digit. It has a 1 in the first digit, which is different from the 0 in the first number. It has a 0 in the second digit, which is different from the 3 in the second number. And so on.

So even though we had a never-ending list of numbers, we showed that there are even more numbers that aren't on the list. This means that some sets of numbers are bigger than others, and that's what Cantor's diagonal argument is all about.