A strong antidiamond principle compatible with

A strong antidiamond principle (⋆c) is shown to be consistent with . This principle can be stated as a “P-ideal dichotomy”: every P-ideal on ω1 (i.e. an ideal that is σ-directed under inclusion modulo finite) either has a closed unbounded subset of ω1 locally inside of it, or else has a stationary subset of ω1 orthogonal to it. We rely on Shelah’s theory of parameterized properness for 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 iteration theory involves definability of forcing notions in third order arithmetic, analogous to Souslin forcing in second order arithmetic.