We explore the theory of illfounded and cyclic proofs for the propositional modal $μ$-calculus. A fine analysis of provability for classical and intuitionistic modal logic provides a novel bridge between finitary, cyclic and illfounded conceptions of proof and re-enforces the importance of two normal form theorems for the logic: guardedness and disjunctiveness.

• 03B45 - Modal logic (including the logic of norms)
• 03F05 - Cut-elimination and normal-form theorems