The paper presents a set-theoretic translation method for polymodal logics that reduces derivability in a large class of propositional polymodal logics to derivability in a very weak first-order set theory Ω. Unlike most existing translation methods, the one we propose applies to any normal complete finitely axiomatizable polymodal logic, regardless of whether it is first-order complete or an explicit semantics is available. The finite axiomatizability of Ω allows one to implement mechanical proof-search procedures via the deduction theorem. Alternatively, more specialized and efficient techniques can be employed. In the last part of the paper, we briefly discuss the application of set T-resolution to support automated derivability in (a suitable extension of) Ω.