Cantor's theorem

Cantor's theorem is like a magic trick that shows us that some things are bigger than we think!

Imagine you have a basket with some apples in it. If you count the apples, you might find that there are 5 of them.

Now imagine you have another basket with oranges in it. You count the oranges and find that there are also 5 of them.

So you might think that the two baskets have the same amount of fruit in them, right? But not so fast!

Cantor's theorem tells us that even though the baskets might have the same number of things in them, they are not actually the same size.

How can that be? Well, imagine if we tried to make a list of all the possible combinations of the apples and oranges in the two baskets. We might have something like this:

- Apple 1 + Orange 1
- Apple 1 + Orange 2
- Apple 1 + Orange 3
- Apple 1 + Orange 4
- Apple 1 + Orange 5
- Apple 2 + Orange 1
- Apple 2 + Orange 2
- Apple 2 + Orange 3
- Apple 2 + Orange 4
- Apple 2 + Orange 5
- Apple 3 + Orange 1
- Apple 3 + Orange 2
- Apple 3 + Orange 3
- Apple 3 + Orange 4
- Apple 3 + Orange 5
- Apple 4 + Orange 1
- Apple 4 + Orange 2
- Apple 4 + Orange 3
- Apple 4 + Orange 4
- Apple 4 + Orange 5
- Apple 5 + Orange 1
- Apple 5 + Orange 2
- Apple 5 + Orange 3
- Apple 5 + Orange 4
- Apple 5 + Orange 5

That's a total of 25 possible combinations. But here's the thing: even though we have 5 apples and 5 oranges, there are actually more possible combinations than that!

Cantor's theorem tells us that the set of all possible combinations is bigger than the set of just the apples or just the oranges. It's like there are more numbers between 0 and 1 than there are whole numbers!

So even though we might think that two sets are the same size, Cantor's theorem shows us that there can be more going on than meets the eye. It's like a cool magic trick that teaches us something new about numbers!