What is Cantor’s theorem?

Cantor’s theorem is about how some infinite sets are actually bigger than others, even though they’re both infinite.

Imagine you have a big bag of marbles, and each marble has a number on it. Now, suppose the numbers go from 1 to forever, that's an infinite set of marbles. But then you get another bag with even more marbles: not just numbers, but pairs of numbers like (1,2), (3,4), (5,6)... and so on. It seems like there should be the same number of marbles in both bags, after all, they’re both infinite!

But here’s where it gets fun: if you try to match each marble from the first bag with one from the second, you’ll always run out of numbers in the first bag before you finish pairing them all. That means the second bag has more marbles, even though both are infinite! This is what Cantor showed: some infinite sets can be bigger than others.

This idea might seem strange, but think about real life: if you have a room full of people and then everyone gets a pair of shoes, there are now more items (shoes) than people, even though both groups are big. Cantor’s theorem is like that, but with infinite numbers instead of shoes!