Forcing axioms and the continuum hypothesis

Abstract

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π2-sentences over the structure (H(ω2), ∈, NSω1), in the sense that its (H(ω2), ∈, NSω1) satisfies every Π2-sentence σ for which (H(ω2), ∈, NSω1) ⊨ σ can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π2-sentences over the structure (H(ω2), ∈, ω1) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $. In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.