How Does Some Infinities ARE Bigger Than Other Infinities (Diagonalization) Work?

Some infinities are bigger than others because you can't match up all the numbers in one group to another, kind of like how you can’t pair every sock with a shoe if there are more shoes than socks.

Think about a hotel with an endless number of rooms, and every room is already full. But then, a new guest shows up! The hotel has infinity rooms, but it still makes space for the new person by asking everyone to move to the next room. That’s how infinite things work: they never run out.

Now picture two hotels like that. One is full of people, and the other is also full of people. But what if one hotel has more people than the other? How can you tell?

Let's say we try to pair up every person from Hotel A with someone in Hotel B. If we can match them all perfectly, one-to-one, then they’re the same size of infinity.

But diagonalization shows that sometimes, no matter how cleverly you try to match them, there will always be at least one extra person left over in the bigger hotel. That means its infinity is bigger than the other one!

It’s like having a never-ending bag of marbles vs. a never-ending bag of jellybeans, both are infinite, but they’re not the same kind of infinite!