A strong antidiamond principle compatible with

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 ω 1 (i.e. an ideal that is σ -directed under inclusion modulo finite) either has a closed unbounded subset of ω 1 locally inside of it, or else has a stationary subset of ω 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.