I can have an alphabet $\mathcal{A}$, a set of axioms $\mathcal{X}$ which are finite strings of $\mathcal{A}$ and a set of rules $\mathcal{R}$.
Every finite strings produced by applying a finite number of time the rules in $\mathcal{R}$ is a theorem.
Is there some theories that studies the axioms and the theorems are infinite strings produced by applying a finite number of time the rules in $\mathcal{R}$?