What should $\aleph_2$ mean?

Question

Just curious, what should $\aleph_2$ mean? I know that $\aleph_1$ is distinct from $\aleph_0$ and $\aleph_1$ is not countable, but what does $\aleph_2$ mean?

Answer 1

$\aleph_0$ is the cardinality of the set of all the finite ordinals, which coincide with our usual interpretation of the natural numbers. $\aleph_1$ is the cardinality of the set of all the countable ordinals.

$\aleph_2$ is the cardinality of the set of all the ordinals whose size is $\leq\aleph_1$ (and in fact equals $\aleph_1$ would suffice).

If we assume the generalized continuum hypothesis, then $\aleph_2=|\mathcal P(\Bbb R)|$.

Answer 2

To approach this in a slightly different manner than Asaf, in general for every ordinal number $\alpha$ we designate by $\aleph_\alpha$ the unique infinite cardinal number $\kappa$ such that the set of all infinite cardinals $< \kappa$ has order type $\kappa$ (according to the usual order). So one may think of $\aleph_\alpha$ is the "$(\alpha+1)^{\text{st}}$ infinite cardinal".

I'm sure you already know that $\aleph_0$ is the smallest (first) infinite cardinal.

$\aleph_1$ is then the least cardinal number greater than $\aleph_0$, the second infinite cardinal, and the first uncountable cardinal.

$\aleph_2$ is the least cardinal number greater than $\aleph_1$. (The third infinite cardinal, and the second uncountable one.)

Ans we can continue like this ad infinitum.