Many countable support iterations of proper forcings preserve Souslin trees

by Mildenberger and Shelah. [MdSh:973]
Annals Pure and Applied Logic, 2014
We show that there are many models of {cov} {ensuremath {M}}= aleph_1 and cof {ensuremath {M}} = aleph_2 in which the club principle holds and there are Souslin trees. The proof consists of the following main steps: item [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, {S})-preserving. A special case of (T,Y, {S})-preserving is preserving the Souslinity of an omega_1-tree. item [2.] We show that some tree-creature forcings from [RoSh:470] satisfy the sufficient condition for one of the strongest games. item [3.] Without the games, we show that some linear creature forcings from [RoSh:470] are (T,Y, {S})-preserving. There are non-Cohen preserving examples. item [4.] For the wider class of non-elementary proper forcings we show that omega-Cohen preserving for certain candidates implies (T,Y, {S})-preserving. item [5.] (+ 1 978)We give a less general but hopefully more easily readable presentation of a result from [Sh:f, Chapter~18, section 3]: If all iterands in a countable support iteration are proper and (T,Y, {S})-preserving, then also the iteration is (T,Y, {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 {{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.

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