In one sentence, what goes on in M is that from the world u, there is an infinite number of finite π-paths of growing length. We have both M, u sat ¬Δπ and M, u sat ¬∞(π*). Yet, for every PDL formula A, we will have both Δπ and ∞(π*) that are satisfied at the equivalence class of u in the model obtained by filtration of M with FL(A). Indeed, the filtration must collapse some states of M and create some loops. Thus, there exists no PDL formula that can express either Δπ or ∞(π*).  and yet, we will have both Δπ and ∞(π*) that are satisfied at the equivalence class of u in any finite filtrate of M. Thus, neither Δπ nor ∞(π*) can be expressed in PDL.