This question comes from an exam and I can't find a nice solution,
Consider the set $A=\{f:\omega \rightarrow \omega |f \space\text{ is injective and }\space f \subseteq S\}$. What are the possible infinite values of $|A|$ given $S \in \mathcal{P}(\omega \times \omega)$?
I can easily get $\aleph_0$ and $2^{\aleph_0}$, but what about any cardinality $\kappa$ such that $\aleph_0 \lt \kappa \lt 2^{\aleph_0}$? My claim is that they can't be obtained, but any attempt to prove it failed. I tried also to show that it is indecidable, but I don't know I to proceed either (and also I don't really think that is the answer).
Thank you for your help
Isn't this precisely the continuum hypothesis? That is, the statements "$|A|$ can only have 1 of two infinite values" and "$|A|$ can have more than two infinite values" are both independent (neither true nor false) in ZFC. Either can be taken as a new axiom to ZFC.