Forcing axioms and the continuum hypothesis

• 成果类型: Article
• 署名作者: Aspero, David; Larson, Paul; Moore, Justin Tatch
• 署名单位: University of East Anglia; University System of Ohio; Miami University; Cornell University
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-013-0089-7
• 发表日期: 2013
• 页码: 1-29
• 关键词: omega(1)

摘要

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for I (2)-sentences over the structure (H(omega (2)), a, NS (omega 1)), in the sense that its (H(omega (2)), a, NS (omega 1)) satisfies every I (2)-sentence sigma for which (H(omega (2)), a, NS (omega 1)) aS sigma can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two I (2)-sentences over the structure (H(omega (2)), a, omega (1)) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies . 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.