Justifying induction on modal μ-formulae

We define a rank function rk, mapping modal μ-formulae on ordinals below ω such that rk(μx.φ) > rk(φ(x)) for n ∈ N, with φ(x) ≡ x and φ(x) ≡ φ[x/φ(x)], and rk(φ) > rk(ψ) for all proper subformulae ψ of φ. The corresponding induction on formulae is thus justified by transfinite induction on ω. We then show that ω is the least such ordinal. Further, we give an algorithm to compute rk(φ) by primitive recursion and show how to get formulae φ of any rank rk(φ) in a uniform way.