Forcing axioms and the continuum hypothesis

David Asperό School of Mathematics, University of East Anglia Paul Larson Department of Mathematics, Miami University Justin Tatch Moore Department of Mathematics, Cornell University

Logic mathscidoc:1701.21001

Acta Mathematica, 210, (1), 1-29, 2010.10
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
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  title={Forcing axioms and the continuum hypothesis},
  author={David Asperό, Paul Larson, and Justin Tatch Moore},
  booktitle={Acta Mathematica},
David Asperό, Paul Larson, and Justin Tatch Moore. Forcing axioms and the continuum hypothesis. 2010. Vol. 210. In Acta Mathematica. pp.1-29.
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