# Sh:973

• Mildenberger, H., & Shelah, S. (2014). Many countable support iterations of proper forcings preserve Souslin trees. Ann. Pure Appl. Logic, 165(2), 573–608.
• Abstract:
We show that there are many models of \rm{cov} {\mathcal M}= \aleph_1 and {\rm cof}{{{\mathcal M}}} = \aleph_2 in which the club principle holds and there are Souslin trees. The proof consists of the following main steps:

1. We give some iterable and some non-iterable conditions on a forcing in terms of games that imply that the forcing is (T,Y,\mathcal{S})-preserving. A special case of (T,Y,\mathcal{S})-preserving is preserving the Souslinity of an \omega_1-tree.

2. We show that some tree-creature forcings from [RoSh:470] satisfy the sufficient condition for one of the strongest games.

3. Without the games, we show that some linear creature forcings from [RoSh:470] are (T,Y,\mathcal{S})-preserving. There are non-Cohen preserving examples.

4. For the wider class of non-elementary proper forcings we show that \omega-Cohen preserving for certain candidates implies (T,Y,\mathcal{S})-preserving.

5. (+ 1 978)We give a less general but hopefully more easily readable presentation of a result from [Sh:f, Chapter 18, §3]: If all iterands in a countable support iteration are proper and (T,Y,\mathcal{S})-preserving, then also the iteration is (T,Y,\mathcal{S})-preserving. This is a presentation of the so-called case A in which a division in forcings that add reals and those who do not is not needed.

In [Mi:clubdistr] we showed: Many proper forcings from [RoSh:470] with finite or countable {\rm{H}}(n) (see Section 2.1) force over a ground model with \diamondsuit_{\omega_1} in a countable support iteration the club principle. After \omega_1 iteration steps the diamond holds anyway.

• Current version: 2013-08-20_11 (44p) published version (36p)
Bib entry
@article{Sh:973,
author = {Mildenberger, Heike and Shelah, Saharon},
title = {{Many countable support iterations of proper forcings preserve Souslin trees}},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {165},
number = {2},
year = {2014},
pages = {573--608},
issn = {0168-0072},
mrnumber = {3129729},
mrclass = {03E05 (03E17 03E35 03E40)},
doi = {10.1016/j.apal.2013.08.002},
note = {\href{https://arxiv.org/abs/1309.0196}{arXiv: 1309.0196}},
arxiv_number = {1309.0196}
}