April 17
Corey Switzer,
CUNY
Specializing Wide Trees Without Adding Reals
An important advancement in iterated forcing was Jensen’s proof that CH does not imply $\diamondsuit$ by iteratively specializing Aronszajn trees with countable levels without adding reals thus producing a model of CH plus 'all Aronszajn trees are special'. This proof was improved by Shelah who developed a general method around the notion of dee-complete forcing. This class (under certain circumstances) can be iterated with countable support and does not add reals. However, neither Jensen's nor Shelah's posets will specialize trees of uncountable width and it remains unclear when one can iteratively specialize wider trees. Indeed a very intriguing example, due to Todorčević, shows that there is always a wide Aronszajn tree which cannot be specialized without adding reals. By contrast the ccc forcing for specializing Aronszajn trees makes no distinction between trees of different widths (but may add many reals). In this talk we will provide a general criteria a wide trees Aronszajn tree can have that implies the existence of a dee-complete poset specializing it. Time permitting we will discuss applications of this forcing to forcing axioms compatible with CH and some open questions related to set theory of the reals.