Journal of Symbolic Logic

Extender Based Forcings

Moti Gitik and Menachem Magidor

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The paper is a continuation of [The SCH revisited]. In $\S 1$ we define a forcing with countably many nice systems. It is used, for example, to construct a model "GCH below $\kappa, c f \kappa = \aleph_0$, and $2^\kappa > \kappa^{+\omega}$" from $0(\kappa) = \kappa^{+\omega}$. In $\S 2$ we define a triangle iteration and use it to construct a model satisfying "$\{\mu \leq \lambda\mid c f \mu = \aleph_0$ and $pp(\mu) > \lambda\}$ is countable for some $\lambda$". The question of whether this is possible was asked by S. Shelah. In $\S 3$ 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 "$c f \kappa = \aleph_0$, GCH below $\kappa, 2^\kappa > \kappa^+$, and $\neg\square^\ast_\kappa$". In $\S 4$ a variation of the forcing of [The SCH revisited, $\S 1$] is defined. It behaves nicely in iteration processes. As an application, we sketch a construction of a model satisfying: "$\kappa$ is a measurable and $2^\kappa \geq \kappa^{+\alpha}$ for some $\alpha, \kappa < c f \alpha < \alpha$" starting with $0(\kappa) = \kappa^{+\alpha}$. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.

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J. Symbolic Logic, Volume 59, Issue 2 (1994), 445-460.

First available in Project Euclid: 6 July 2007

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Gitik, Moti; Magidor, Menachem. Extender Based Forcings. J. Symbolic Logic 59 (1994), no. 2, 445--460.

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