Can an intersection of two context-free languages be an undecidable language? I'm trying to prove that

$\exists L_1, L_2 : L_1$ and $L_2$ are context-free languages $\land\;L_1 \cap L_2 = L_3$ is an undecidable language.



I know that context-free languages are not closed under intersection.

This means that I can produce an $L_3$, which is undecidable.

An example would be $L_1 = \{a^n | n \in \mathbb{N}\} \cap L_2 = \{0\} = \emptyset$.


Is this a correct proof?
If not, how can I prove this theorem?
Is the empty language decidable?
Context-free languages are decidable, and decidable languages are
closed under intersection.

So, though the intersection of two CF languages may not be CF, it is decidable.

Remarks on your example:


$\emptyset=\{\}\neq$ $\{0\}$
$L_1\cap\emptyset=\emptyset$  which is context-free.
You cannot prove your claim, because it is wrong
the empty language is decidable: the answer is always "no, this string is not in the empty set".