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Logicmathscidoc: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
```@inproceedings{david2010forcing,
title={Forcing axioms and the continuum hypothesis},
author={David Asperό, Paul Larson, and Justin Tatch Moore},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400077378765},
booktitle={Acta Mathematica},
volume={210},
number={1},
pages={1-29},
year={2010},
}
```
David Asperό, Paul Larson, and Justin Tatch Moore. Forcing axioms and the continuum hypothesis. 2010. Vol. 210. In Acta Mathematica. pp.1-29. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400077378765.