Acta Mathematica

Forcing axioms and the continuum hypothesis

David Asperό, Paul Larson, and Justin Tatch Moore

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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 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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E35: Consistency and independence results
Secondary: 03E50: Continuum hypothesis and Martin's axiom [See also 03E57] 03E57: Generic absoluteness and forcing axioms [See also 03E50]

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

2013 © Institut Mittag-Leffler


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.

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