Abstract

Let T be an $\omega_{1}-Souslin$ tree. We show the property of forcing notions; "is $\lbrace\omega_{1}\rbrace-semi-proper$ and preserves T" is preserved by a new kind of revised countable support iteration of arbitrary length. As an application we have a forcing axiom which is compatible with the existence of an $\omega_{1}-Souslin$ tree for preorders as wide as possible