Schlindwein, C., Suslin's hypothesis does not imply stationary antichains, Annals of Pure and Applied Logic 64 153–167. Shelah has shown that Suslin's hypothesis does not imply every Aronszajn tree is special. We improve this result by constructing a model of Suslin's hypothesis in which some Aronszajn tree has no antichain whose levels constitute a stationary set. The main point is a new preservation theorem, the proof of which illustrates the usefulness of certain ideas in [8, Section 1].

There are two versions of the Proper Iteration Lemma. The stronger version can be used to give simpler proofs of iteration theorems . In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ωω-bounding property.

We give a simplified treatment of revised countable support (RCS) forcing iterations, previously considered by Shelah (see [Sh, Chap. X]). In particular we prove the fundamental theorem of semi-proper forcing, which is due to Shelah: any RCS iteration of semi-proper posets is semi-proper.

In this paper, we give details of results of Shelah concerning iterated Namba forcing over a ground model of CH and iteration of P[W] where W is a stationary subset of ω 2 concentrating on points of countable cofinality.

There are two versions of the Proper Iteration Lemma. The stronger (but less well‐known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ωω‐bounding property. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

We build a model in which the continuum hypothesis and Suslin's hypothesis are true, yet there is an Aronszajn tree with no stationary antichain.

We present an exposition of much of Sections VI.3 and XVIII.3 from Shelah's book Proper and Improper Forcing. This covers numerous preservation theorems for countable support iterations of proper forcing, including preservation of the property “no new random reals over V ”, the property “reals of the ground model form a non-meager set”, the property “every dense open set contains a dense open set of the ground model”, and preservation theorems related to the weak bounding property, the weak ωω -bounding (...) property, and the property “the set of reals of the ground model has positive outer measure”. (shrink)

We present an exposition of Section VI.1 and most of Section VI.2 from Shelah’s book Proper and Improper Forcing. These sections offer proofs of the preservation under countable support iteration of proper forcing of various properties, including proofs that ωω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega^\omega}$$\end{document} -bounding, the Sacks property, the Laver property, and the P-point property are preserved by countable support iteration of proper forcing.