The Journal of Symbolic Logic

Research Article

Extender based forcings

Moti Gitika1 and Menachem Magidora2

a1 Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel, E-mail: gitik@math.tan.ac.il

a2 Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel, E-mail: menachem@math.huji.ac.il

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.

(Received October 29 1991)

(Revised February 15 1993)