What are infinite cardinalities?

There are different kinds of infinite things, some are bigger than others, just like how a big bag of jellybeans is bigger than a small one.

Imagine you have two jars: one with countable jellybeans and another with uncountable jellybeans. The countable jar is like the jellybeans you can count one by one, 1, 2, 3, all the way up. That’s like how we count numbers on a number line.

Now the uncountable jar is different. It's like having jellybeans that keep appearing in between every other jellybean, no matter how much you count, there are always more to go. This is like infinite cardinalities, different sizes of infinity!

If you have a bag with numbers like 1, 2, 3, ..., it's the same size as a bag with just even numbers: 2, 4, 6, ... You can match them one-to-one. That’s a countable infinity.

But if you try to count all the points on a line, every little spot between the jellybeans, that’s an uncountable infinity. No matter how cleverly you pair them up, there are always more spots than numbers.

So, even though both are infinite, one is bigger than the other, like having two bags of jellybeans, but one has way more! There are different kinds of infinite things, some are bigger than others, just like how a big bag of jellybeans is bigger than a small one.

Imagine you have two jars: one with countable jellybeans and another with uncountable jellybeans. The countable jar is like the jellybeans you can count one by one, 1, 2, 3, all the way up. That’s like how we count numbers on a number line.

Now the uncountable jar is different. It's like having jellybeans that keep appearing in between every other jellybean, no matter how much you count, there are always more to go. This is like infinite cardinalities, different sizes of infinity!