  We axiomatize the provability logic of HA and show that it is decidable.
Moreover we axiomatize the preservativity and relative admissibility for
several modal logics extending iK4. As a main tool, we also provide some sort
of semantics, called provability semantics, for modal logics extending iGL,
which is a mixture of usual Kripke semantics and provability in propositional
modal logics.
