  In the final chain of the countable powerset functor, we show that the set at
index $\omega_1$, regarded as a transition system, is not strongly extensional
because it contains a "ghost" element that has no successor even though its
component at each successor index is inhabited. The method, adapted from a
construction of Forti and Honsell, also gives ghosts at larger ordinals in the
final chain of other subfunctors of the powerset functor. This leads to a
precise description of which sets in these final chains are strongly
extensional.
