## Journal of Symbolic Logic

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

### Extender Based Forcings

Moti Gitik and Menachem Magidor

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 6 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1183744489

**Mathematical Reviews number (MathSciNet)**

MR1276624

**Zentralblatt MATH identifier**

0820.03032

**JSTOR**

links.jstor.org

#### Citation

Gitik, Moti; Magidor, Menachem. Extender Based Forcings. J. Symbolic Logic 59 (1994), no. 2, 445--460. https://projecteuclid.org/euclid.jsl/1183744489