Can we use Axiomitization to decide CTL*?

As the title says. There are axioms for CTL*, but can we use them to decide whether a formula is satisfiable? We can use the trick $$p$$ is satisfiable $$\iff$$ $$\lnot p$$ is not valid. My problem is that the formulas arent growing monotonically, so we would never know when to quit.