Extender based forcings

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 “{μ ≤ λ∣c 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.