Justifying induction on modal μ-formulae

We define a rank function for formulae of the propositional modal μ-calculus such that the rank of a fixed point is strictly bigger than the rank of any of its finite approximations. A rank function of this kind is needed, for instance, to establish the collapse of the modal μ-hierarchy over transitive transition systems. We show that the range of the rank function is ωω. Further we establish that the rank is computable by primitive recursion, which gives us a uniform method to generate formulae of arbitrary rank below ωω.

A note on the theory SID<ω of stratified induction

We introduce and analyse a theory of finitely stratified general inductive definitions over the natural numbers, inline image, and establish its proof theoretic ordinal, inline image. The definition of inline image bears some similarities with Leivant's ramified theories for finitary inductive definitions.