This paper presents a sound and complete logical system whose atomic sentences are the equalities of recursive terms involving sets. There are two interpretations of this language: one makes use of non-wellfounded sets with finite transitive closure, and the other uses pointed finite graphs modulo bisimulation. Our logical system is a sequent-style deduction system. The main axioms and inference rules come from the
-proof system from , including the Recursion Inference Rule (but an additional axiom is needed), and also axioms corresponding to the extensionality axiom of set theory.