How many mappings from $\mathbb C$ to $\mathbb C$ are there?

Question

I read this question a moment ago, and am now wondering how many possible mappings there are between $\mathbb C$ and $\mathbb C$. Correct me if I'm wrong but I seem to remember the cardinality of the complex numbers being $\aleph_1$ (I can't find this after a brief google search) thus the number of mappings from the complex numbers to the complex numbers would be ${}^2\aleph_1$. Is there a symbol for this value? Is this value equivalent to any other non-trivial expression involving infinite and transfinite numbers?

Sorry if this question seems uneducated, I've only read about the aleph numbers several times and can't remember, or find much on them now.

Answer 1

The cardinality of complex numbers is $\beth_1:=2^{\aleph_0}$. Thus, the cardinality of the set of maps $\Bbb C\to\Bbb C$ is $$\beth_1^{\beth_1}=\left(2^{\aleph_0}\right)^{2^{\aleph_0}}=2^{\aleph_0\cdot 2^{\aleph_0}}=2^{2^{\aleph_0}}=2^{\beth_1}$$

The latter is, by definition, called $\beth_2$ (see "beth numbers").

Values of $\alpha$ such that any of the aforementioned $\beth$ is $\aleph_\alpha$ cannot be determined exactly and, in fact, they may be chosen with some degree of arbitrariness without losing consistency with ZFC (this is the so-called "generalized continuum hypothesis").