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Extender based forcings

  • Moti Gitik (a1) and Menachem Magidor (a2)

The paper is a continuation of [The SCH revisited], In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model “GCH below κ, c f κ = ℵ0, and 2κ > κ+ω from 0(κ) = κ+ω. In §2 we define a triangle iteration and use it to construct a model satisfying “{μλc f μ = ℵ0 and pp(μ) > λ} is countable for some λ”. The question of whether this is possible was asked by S. Shelah. In §3 a forcing for blowing the power of a singular cardinal without collapsing cardinals or adding new bounded subsets is presented. Answering a question of H. Woodin, we show that it is consistent to have “c f κ = ℵ0. GCH below κ, 2κ > κ+, and ”. In §4 a variation of the forcing of [The SCH revisited, §1] is defined. It behaves nicely in iteration processes. As an application, we sketch a construction of a model satisfying:

κ is a measurable and 2κκ+α for some α, κ < c f α < α” starting with 0(κ) = κ+α. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.

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[G1] M. Gitik , Changing cofinalities and the nonstationary ideal, Israel Journal of Mathematics, vol. 56 (1986), pp. 28314.

[G2] M. Gitik , The negation of SCH from 0(κ) = κ++, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 209234.

[G3] M. Gitik , On measurable cardinals violating the continuum hypothesis, Annals of Pure and Applied Logic, vol. 63 (1993>, pp. 227240.

[Jen] R. Jensen , The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229309.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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