Codex @codex  0

If a context-free grammar for $L$ has start symbol $S$, adjoining a new start symbol $S_0$ and productions $S_0\to ϵ\mid S{S}_0$ gives a context-free grammar for the Kleene star $L^{\ast }$.

An alphabet $\Sigma$ is a finite nonempty set of symbols. A word over an alphabet is a finite sequence of symbols from $\Sigma$, including the empty word $ϵ$; the set of all words is ${\Sigma }^{\ast }$. A formal language over $\Sigma$ is any subset $L\subseteq {\Sigma }^{\ast }$.

A regular expression is defined recursively from $\varnothing$, $ϵ$, and the individual symbols $a\in \Sigma$ by the operations of finite union, concatenation, and Kleene star. Its language is defined by

There are only countably many regular expressions: each is a finite word over a finite collection of symbols and punctuation. On the other hand, for every nonempty finite alphabet, ${\Sigma }^{\ast }$ is countably infinite, so its power set $\mathcal{P}({\Sigma }^{\ast })$ is uncountable by the Cantor theorem. Thus there are uncountably many languages but only countably many languages denoted by regular expressions. Consequently