  Let M be a model of Peano Arithmetic that is countably generated over an
exponentially closed cut I. We characterize those sets X of subsets of I for
which there is a finitely (or countably) generated cofinal extension N of M
such that I is the Greatest Common Initial Segment of M and N and X is the set
of subsets of I coded by the extension. We also characterize such X for which
the extension N can be, in addition, one of the following: non-filling;
filling; n-filling.
