Modal Frame Correspondence Generalized Johan van Benthem Abstract: We show how first-order modal frame correspondence methods can be generalized to deal with axioms like Löb's, provided we compute minimal substitutions in a fixed-point extension of first-order logic. Then we restore the harmony by also considering frame correspondence for modal fixed-point languages like PDL or the MU-calculus. We show how the latter leads to elegant and quite workable proof systems, and hence to versions of modal logic and provability logic that may be at least as natural in many ways. This paper will appear in a special issue of "Studia Logica" devoted to the memory of Wim Blok. Keywords: modal logic; frame correspondence; ; fixed-point logic; substitution algorithm; ; minimal predicates; provability logic; ; Löb's axiom