Language containing all unambiguous grammars

Question

Suppose $L$ is the language of the unambiguous grammars. That is, a sentence $w\in{}L$ if it is a string that describes an unambiguous context-free grammar.

Considering that deciding whether a context-free grammar is ambiguous is non-decidable, would it be correct to say that $L$ exists but is not recursively enumerable?

Answer 1

The language $L$ clearly exists — you just defined it!

It’s not hard to check that your language is co-r.e., that is, you can enumerate ambiguous grammars. Since it’s not recursive, it cannot be r.e.