Unprovability of the logical characterization of bisimulation

We quickly review labelled Markov processes (LMP) and provide a counterexample showing that in general measurable spaces, event bisimilarity and state bisimilarity differ in LMP. This shows that the Hennessy-Milner logic proposed by Desharnais does not characterize state bisimulation in non-analytic measurable spaces. Furthermore we show that, under current foundations of Mathematics, such logical characterization is unprovable for spaces that are projections of a coanalytic set. Underlying this construction there is a proof that stationary Markov processes over general measurable spaces do not have semi-pullbacks.