Extender Based Forcings
Gitik, Moti ; Magidor, Menachem
J. Symbolic Logic, Tome 59 (1994) no. 1, p. 445-460 / Harvested from Project Euclid
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.
Publié le : 1994-06-14
Classification: 
@article{1183744489,
     author = {Gitik, Moti and Magidor, Menachem},
     title = {Extender Based Forcings},
     journal = {J. Symbolic Logic},
     volume = {59},
     number = {1},
     year = {1994},
     pages = { 445-460},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1183744489}
}
Gitik, Moti; Magidor, Menachem. Extender Based Forcings. J. Symbolic Logic, Tome 59 (1994) no. 1, pp.  445-460. http://gdmltest.u-ga.fr/item/1183744489/