Iterated souslin forcing, the principles ⋄(E) and a generalisation of the axiom SAD

Abstract

The axiom SAD was introduced in our paper with Avraham and Shelah [1]. It is a Martin’s Axiom type of principle, having some of the consequences of MA plus\(2^{\aleph _0 } > \aleph _1 \), but nonetheless provably consistent with GCH. In [1] it was shown to be consistent (with GCH) and used to demonstrate the consistency with CH of some known consequences of MA +\(2^{\aleph _0 } > \aleph _1 \). In particular, SAD implies the negation of Jensen’s ⋄ principle. In this paper we present a generalisation of SAD, let us call it SAD(E), whereE will be an arbitrary stationary subset ofω1, and show that although SAD(E) implies the negation of ⋄(E), it is consistent with ⋄. SAD(E) resembles the axiom SA of Shelah, described in our survey article [2], and indeed is a sort of blending of the two principles SA and SAD. (In particular, Shelah proved that SA is consistent with ⋄ but implies the failure of some ⋄(E).) Our proof (of the consistency of SAD(E) with ⋄) will be of interest to forcing enthusiasts, since it shows that iterated Souslin forcingcan distinguish between different stationary sets (it was previously thought that this was not the case), and can indeed be used to establish the non-provability of the principles ⋄(E) from ⋄ alone.