- Alphabet $Σ$: finite, non-empty set
- Language: subset of $Σ^*$
- Grammar: Unrestricted grammar (Chomsky Type 0)
- Language of a grammar: all words that can be produced by applying $P$ multiple times, starting from $S$
Grammars are finite, therefore there are only countable infinite of them. But there are uncountably infinite many languages. Each grammar can only describe one language. Therefore, there are languages without grammars.
Can you give an example for such a language without grammar?
I searched the internet, but strangely, I could not even find the question in context of formal language.