$\aleph_1$ and the Modal $\mu$-Calculus

International audienceFor a regular cardinal κ, a formula of the modal µ-calculus is κ-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of κ-directed sets. We define the fragment C ℵ 1 (x) of the modal µ-calculus and prove that all the formulas in this fragment are ℵ 1-continuous. For each formula φ(x) of the modal µ-calculus, we construct a formula ψ(x) ∈ C ℵ 1 (x) such that φ(x) is κ-continuous, for some κ, if and only if φ(x) is equivalent to ψ(x). Consequently, we prove that (i) the problem whether a formula is κ-continuous for some κ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment C ℵ 0 (x) studied by Fontaine and the fragment C ℵ 1 (x). We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal µ-calculus. An ordinal α is the closure ordinal of a formula φ(x) if its interpretation on every model converges to its least fixed-point in at most α steps and if there is a model where the convergence occurs exactly in α steps. We prove that ω 1 , the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, ω, ω 1 by using the binary operator symbol + gives rise to a closure ordinal