Shelah's work on non-semi-proper iterations. I (Q944227)

The author reworks results of Shelah concerning non-semi-proper iterations. It is shown that any forcing notion satisfying a certain condition preserves \(\omega_1\) (and, assuming CH, adds no reals). Namba forcing satisfies the condition, and for each stationary \(W\subseteq \{ \alpha < \omega_2 : \text{cf} (\alpha) = \omega\}\), so does \(P[W]\) (\(P[W]\) is the set of all countable closed subsets of \(W\), ordered by reverse end-extension). The main result asserts that the condition is preserved for appropriate RCS iterations.