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Extender based forcings

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

Abstract

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 2 κ > 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, 2 κ > 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 2 κ κ +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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References

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[G1] Gitik, M., Changing cofinalities and the nonstationary ideal, Israel Journal of Mathematics, vol. 56 (1986), pp. 28314.
[G2] Gitik, M., The negation of SCH from 0(κ) = κ ++ , Annals of Pure and Applied Logic, vol. 43 (1989), pp. 209234.
[G3] Gitik, M., On measurable cardinals violating the continuum hypothesis, Annals of Pure and Applied Logic, vol. 63 (1993>, pp. 227240.
[G-M] Gitik, M. and Magidor, M., The SCH revisited.
[Jen] Jensen, R., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229309.
[Sh] Shelah, S., On successors of singular cardinals, Logic Colloquium, 78 (Boffa, M., van Dallen, D., and McAloon, K., editors), North-Holland, Amsterdam, pp. 357380.
[Sh1] Shelah, S., Cardinal arithmetic , a forthcoming book.
[Sh2] Shelah, S., Cardinal arithmetic for skeptics, Bulletin, American Mathematical Society, Providence, Rhode Island (to appear).

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