More Notions of Forcing Add a Souslin Tree

Abstract

An ${\aleph }_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ${\aleph }_1$-Souslin tree.

In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a ${\lambda }^{+}$-Souslin tree. This class includes Prikry, Magidor, and Radin forcing.