${\mathrm{\aleph }}_1$ 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 $C_{{\mathrm{\aleph }}_1}(x)$ of the modal $\mu$-calculus and prove that all the formulas in this fragment are ${\mathrm{\aleph }}_1$-continuous. For each formula $\varphi (x)$ of the modal $\mu$-calculus, we construct a formula $\psi(x) \in C_{\aleph_1 }(x)$suchthat$\phi(x)$is$\kappa$-continuous,forsome$\kappa$,ifandonlyif$\phi(x)$isequivalentto$\psi(x)$. 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 $C_{{\mathrm{\aleph }}_0}(x)$ studied by Fontaine and the fragment $C_{{\mathrm{\aleph }}_1}(x)$. 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 $\varphi (x)$ 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.