There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou’s probabilistic bisimilarity, whereas the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal {O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the Zhou ordinal of $\mathbb {S}$, $\mathfrak {Z}(\mathbb {S})$. When $\mathfrak {Z}(\mathbb {S})=0$, $\mathbb {S}$ satisfies the Hennessy–Milner property. The second author proved the existence of an LMP $\mathbb {S}$ with $\mathfrak {Z}(\mathbb {S}) \geq 1$ and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb {S}$ over separable metrizable spaces having arbitrary large countable $\mathfrak {Z}(\mathbb {S})$ and that it is consistent with the axioms of $\mathit {ZFC}$ that there is such a process with an uncountable Zhou ordinal.

We investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega ,\omega _1}$ -game under ZFC+MA+ $\neg $ CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega _1}$ -game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.

We prove that Voiculescu’s noncommutative version of the Weyl-von Neumann Theorem can be extended to all unital, separably representable $\mathrm {C}^\ast $ -algebras whose density character is strictly smaller than the (uncountable) cardinal invariant $\mathfrak {p}$ . We show moreover that Voiculescu’s Theorem consistently fails for $\mathrm {C}^\ast $ -algebras of larger density character.

The shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.

Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).

In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.

We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).