  A strong antidiamond principle (*c) is shown to be consistent with CH. This
principle can be stated as a "P-ideal dichotomy": every P-ideal on omega-1
(i.e. an ideal that is sigma-directed under inclusion modulo finite) either has
a closed unbounded subset of omega-1 locally inside of it, or else has a
stationary subset of omega-1 orthogonal to it. We rely on Shelah's theory of
parameterized properness for NNR iterations, and make a contribution to the
theory with a method of constructing the properness parameter simultaneously
with the iteration. Our handling of the application of the NNR iteration theory
involves definability of forcing notions in third order arithmetic, analogous
to Souslin forcing in second order arithmetic.
