On Semi-Proper Forcing

Abstract

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some iterations) are preserved and it includes some forcing which changes the cofinality of a regular cardinal > ℵ ₁ to ℵ ₀. So, using the right iterations, we can iterate such forcing without collapsing ℵ ₁. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.:

• (1) for every S ⊆ ℵ ₂, S or ℵ ₂ − S contains a closed copy of ω ₁,

• (2) there is a normal precipitous filter D on ℵ ₂, {δ < ℵ ₂: cf δ = ℵ ₀} ∈ D,

• (3) for every A ⊆ N₂, {δ < ℵ ₂: cf δ = ℵ ₀, δ is regular in L(δ ∩ A)} is stationary.