Title: $\aleph_1$ and the modal $μ$-calculus,
Abstract: 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.