## Acta Mathematica

### 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.

#### Article information

Source
Acta Math., Volume 210, Number 1 (2013), 1-29.

Dates
Revised: 8 December 2011
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892671

Digital Object Identifier
doi:10.1007/s11511-013-0089-7

Mathematical Reviews number (MathSciNet)
MR3037610

Zentralblatt MATH identifier
1312.03031

Rights

#### Citation

Asperό, David; Larson, Paul; Moore, Justin Tatch. Forcing axioms and the continuum hypothesis. Acta Math. 210 (2013), no. 1, 1--29. doi:10.1007/s11511-013-0089-7. https://projecteuclid.org/euclid.acta/1485892671

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