Are there countably many infinities?

Question

$\aleph_0$, $\aleph_1, \aleph_2$ and so on are indexed by a natural number so shouldn't there be countably many infinities?

Answer 1

After you exhaust all the $\aleph_n$, you still have $\aleph_\omega$ which is a cardinality larger than all the $\aleph_n$'s. Then you have $\aleph_{\omega+n}$, for integer $n$, and so on.

And you have many many many more cardinals. In fact, for every ordinal $\alpha$ you have cardinal $\aleph_\alpha$. Since there are uncountable ordinals, there are at least uncountably many cardinals. But in fact, the collection of cardinals does not make a set, it is a proper class.