  Arbitrary Public Announcement Logic with Common Knowledge (APALC) is an
extension of Public Announcement Logic with common knowledge modality and
quantifiers over announcements. We show that the satisfiability problem of
APALC on S5-models, as well as that of two other related logics with
quantification and common knowledge, is $\Sigma^1_1$-hard. This implies that
neither the validities nor the satisfiable formulas of APALC are recursively
enumerable. Which, in turn, implies that APALC is not finitely axiomatisable.
