一句话：证明模态 μ-演算的连续性片段仅有 ℵ0 与 ℵ1 两类，并刻画闭包序数。
本文研究模态 μ-演算公式在变量上的 κ-连续性：公式解释为单变量函数时单调，并保持 κ-有向并。作者定义片段 C_{ℵ1}(x)，证明其中公式均 ℵ1-连续；进而对任意公式 φ(x) 构造 ψ(x)∈C_{ℵ1}(x)，使 φ 对某正则基数 κ 连续当且仅当 φ 等价于 ψ。由此证明“某 κ 连续”可判定，并在等价意义下说明由连续性确定的片段只有 Fontaine 的 C_{ℵ0}(x) 和新的 C_{ℵ1}(x)。最后应用于闭包序数，证明 ω1 是闭包序数，闭包序数对序数和封闭，因此由 0、1、ω、ω1 经 + 构造的表达式均为闭包序数。
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.