How many infinities are there?

Blog » Is there more than one infinity?
Yes.

Are some infinities larger than others?
Yes.

Is there a largest infinity?
No.

Does that mean that there are infinitely many infinities?
Yes.

So... which infinity is the number of infinities? How many infinities are there?
There are two types of number. There are ordinal numbers and there are cardinal numbers. The finite ordinals and the finite cardinals are exactly the same. The infinite ordinals and the infinite cardinals are different from one another. Which of the two would you like me to settle first?

How many ordinals are there?
There are infinitely many ordinals.

Okay... but which infinity?
"How many" is a tricky concept in set theory when we start dealing with infinite sets. What you're actually trying to ask is:

What is the cardinality of the set of all ordinals?
Answer: There is no set of all ordinals.
The ordinals form a sequence. Each ordinal is defined in terms of all the ordinals that are smaller than it. ...