What is Cantor’s hierarchy?

Cantor’s hierarchy is like counting infinite numbers, but not just one kind of infinite, many kinds, each bigger than the last.

Imagine you have a bag full of jellybeans. You can count them all, that's finite. But now imagine a bag that never ends, it has an infinite number of jellybeans. That’s like the first step in Cantor’s hierarchy: counting things one by one, forever and ever.

Now think about two bags of jellybeans, both infinite. But what if one bag is so big that even if you matched every jellybean from one bag to the other, there would still be extra jellybeans left over? That's like moving up a level in Cantor’s hierarchy: some infinities are bigger than others.

At first, we count numbers like 1, 2, 3, that’s counting whole numbers, or natural numbers. Then we go to fractions, like halves, thirds, and even tiny pieces of jellybeans. That's a bigger infinity called the real numbers.

And if you keep going, you can find even more kinds of infinite numbers, each one is like a new level in a never-ending candy factory! Cantor’s hierarchy is like counting infinite numbers, but not just one kind of infinite, many kinds, each bigger than the last.

Imagine you have a bag full of jellybeans. You can count them all, that's finite. But now imagine a bag that never ends, it has an infinite number of jellybeans. That’s like the first step in Cantor’s hierarchy: counting things one by one, forever and ever.

Now think about two bags of jellybeans, both infinite. But what if one bag is so big that even if you matched every jellybean from one bag to the other, there would still be extra jellybeans left over? That's like moving up a level in Cantor’s hierarchy: some infinities are bigger than others.