Are the number of infinities countable or uncountable?

Question

We have $\aleph_0$, $\aleph_1$, and so on. But why wouldn't there be an $\aleph_\pi$, for example? What is the proof that the types of infinities are a countable set?

Answer 1

The cardinalities are $\aleph_\alpha$ where $\alpha$ is an ordinal. So you can have $\aleph_\omega$ (which is the least cardinality larger than $\aleph_0,\aleph_1,\aleph_2,\ldots)$ or even $\aleph_{\omega_1}$ or more. So there are uncountably many, but not because they are indexed by reals, but because there is a proper class of them (i.e. too many to be a set of any size, let alone a countable one).