Specializing Aronszajn Trees and Preserving Some Weak Diamonds

Abstract

We show that ◊(ℝ, , ∈) together with CH and “all Aronszajn trees are special” is consistent relative to ZFC. The weak diamond for the covering relation of Lebesgue null sets was the only weak diamond in the Cichoń diagramme for relations whose consistency together with “all Aronszajn trees are special” was not yet settled. Our forcing proof gives also new proofs to the known consistencies of several other weak diamonds stemming from the Cichoń diagramme together with “all Aronszajn trees are special” and CH. The main part of our work is an application [Shelah, Proper and Improper Forcing, Springer-Velag, 1998, Chapter V, §§ 1–7] for a special completeness system, such that we have a genericity game. Thus we show new preservation properties of the known forcings.