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2013 Forcing axioms and the continuum hypothesis
David Asperό, Paul Larson, Justin Tatch Moore
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Acta Math. 210(1): 1-29 (2013). DOI: 10.1007/s11511-013-0089-7

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.

Citation

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David Asperό. Paul Larson. Justin Tatch Moore. "Forcing axioms and the continuum hypothesis." Acta Math. 210 (1) 1 - 29, 2013. https://doi.org/10.1007/s11511-013-0089-7

Information

Received: 14 October 2010; Revised: 8 December 2011; Published: 2013
First available in Project Euclid: 31 January 2017

zbMATH: 1312.03031
MathSciNet: MR3037610
Digital Object Identifier: 10.1007/s11511-013-0089-7

Subjects:
Primary: 03E35
Secondary: 03E50 , 03E57

Keywords: $Π_2$ maximality , Continuum hypothesis , Forcing axiom , Iterated forcing , Martin’s maximum , Proper forcing axiom

Rights: 2013 © Institut Mittag-Leffler

Vol.210 • No. 1 • 2013
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