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Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π_{2}-sentences over the structure ($H$($ω$_{2}), ∈, NS_{$ω$1}), in the sense that its ($H$($ω$_{2}), ∈, NS_{$ω$1}) satisfies every Π_{2}-sentence$σ$for which ($H$($ω$_{2}), ∈, NS_{$ω$1}) ⊨$σ$can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π_{2}-sentences over the structure ($H$($ω$_{2}), ∈,$ω$_{1}) 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.

Continuum hypothesis; Iterated forcing; Forcing axiom; Martin’s maximum; Π; Proper forcing axiom