Aleph1 and the Modal mu-Calculus

For a regular cardinal $\kappa$, a formula of the modal $\mu$-calculus is $\kappa$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $\kappa$-directed sets. We define the fragment C $\aleph 1 (x) of the modal $\mu$-calculus and prove that all the formulas in this fragment are $\aleph 1-continuous. For each formula $\phi$(x) of the modal $\mu$-calculus, we construct a formula $\psi$(x) $\in$ C $\aleph 1 (x) such that $\phi$(x) is $\kappa$-continuous, for some $\kappa$, if and only if $\phi$(x) is equivalent to $\psi$(x). Consequently, we prove that (i) the problem whether a formula is $\kappa$-continuous for some $\kappa$ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment C $\aleph 0 (x) studied by Fontaine and the fragment C $\aleph 1 (x). We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal $\mu$-calculus. An ordinal $\alpha$ is the closure ordinal of a formula $\phi$(x) if its interpretation on every model converges to its least fixed-point in at most $\alpha$ steps and if there is a model where the convergence occurs exactly in $\alpha$ steps. We prove that $\omega$ 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, $\omega$, $\omega$ 1 by using the binary operator symbol + gives rise to a closure ordinal.