The monadic theory and the "next world"

Suppose that V is a model of ZFC and U ∈ V is a topological space or a richer structure for which it makes sense to speak about the monadic theory. Let B be the Boolean algebra of regular open subsets of U. If the monadic theory of U allows one to speak in some sense about a family of κ everywhere dense and almost disjoint sets, then the second-order V ^B-theory of κ{script} is interpretable in the monadic V-theory of U; this is our Interpretation Theorem. Applying the Interpretation Theorem we strengthen some previous results on complexity of the monadic theories of the real line and some other topological spaces and linear orders. Here are our results about the real line. Let r be a Cohen real over V. The second-order V[r]-theory of א₀ is interpretable in the monadic V-theory of the real line. If CH holds in V then the second-order V[r]-theory of the real line is interpretable in the monadic V-theory of the real line.