Iterated forcing and normal ideals onω 1

Abstract

We prove that suitable iteration does not collapse ℵ1 [and does not add reals], i.e., that in such iteration, certain sealing of maximal antichains of stationary subsets ofω 1 is allowed. As an application, e.g., we prove from supercompact hypotheses, mainly, the consistency of: ZFC + GCH + “for some stationary setS ⊆ω 1, {ie345-1}(ω 1)/(D ω 1 +S) is the Levy algebra” (i.e., the complete Boolean Algebra corresponding to the Levy collapse Levy (ℵ0,<ℵ2) (and we can add “a variant of PFA”) and the consistency of the same, with “Ulam property” replacing “Levy algebra”). The paper assumes no specialized knowledge (if you agree to believe in the semi-properness iteration theorem and RCS iteration).