  Modal logic has been useful in clarifying our understanding of central results concerning provability in the foundations of mathematics (Boolos, 1993). Provability logics are systems where the propositional variables \(p, q, r\), etc. range over formulas of some mathematical system, for example Peano’s system \(\mathbf{PA}\) for arithmetic. (The system chosen for mathematics might vary, but assume it is \(\mathbf{PA}\) for this discussion.) Gödel showed that arithmetic has strong expressive powers. Using code numbers for arithmetic sentences, he was able to demonstrate a correspondence between sentences of mathematics and facts about which sentences are and are not provable in \(\mathbf{PA}\). For example, he showed there there is a sentence \(C\) that is true just in case no contradiction is provable in \(\mathbf{PA}\) and there is a sentence \(G\) (the famous Gödel sentence) that is true just in case it is not provable in \(\mathbf{PA}\).