Show, that there are only countably infinite context-free languages $L_{\text{CFL}}$ over the alphabet $\Sigma=\{a,b\}$.
I know, that I need to find a bijection $f:\mathbb{N}\to L_{\text{CFL}}$ in order to prove, that a set is countably infinite. Unfortunately I don't know how to apply it to that problem. I can't seem to find a bijection, that satisfies a mapping from $\mathbb{N}\to L_{\text{CFL}}$. Can you give me some hints, please?
You don't have to give an explicit bijection with $\Bbb N$. It's enough to realise that there are at most countably many context-free grammars over a fixed alphabet: any CFG can be decribed by a finite tuple of finite things and so you can apply the fact that there are only countably many finite subsets of $\Bbb N$ for instance, and the product of finitely many at most countable sets is at most countable, etc, and ditto for countable unions. Use facts about countable sets... If you feel creative you can create a sort of Gödel-numbering of them, etc.
You can also bound the number of push-down automata, if you prefer, in a similar way.