Abstract

Abstract: In the last years there has been a second boom of the technique of forcing with side conditions (see for instance the recent works of Asper’{o}-Mota, Krueger and Neeman describing three different perspectives of this technique). The first boom took place in the 1980s when Todorcevic discovered a method of forcing in which elementary substructures are included in the conditions of a forcing poset to ensure that the forcing poset preserves cardinals. More than twenty years later, Friedman and Mitchell independently took the first step in generalizing the method from adding small (of size at most the first uncountable cardinal) generic objects to adding larger objects by defining forcing posets with finite conditions for adding a club subset on the second uncountable cardinal. However, neither of these results show how to force (with side conditions together with another finite set of objects) the existence of such a large object together with the continuum being small. In the first part of this talk I will discuss new results in this area. This is joint work with John Krueger improving the symmetric CH preservation argument previously made by Asper’{o} and Mota. In the second part of this talk I will use generalized symmetric systems in order to prove that, for each regular cardinal k, there is a poset $P_k$ forcing the existence of a (k,k++)-superatomic boolean algebra. This is joint work with William Weiss inspired in an unpublished note from September 2009 where Asper’{o} and Bagaria introduced the forcing $P_{omega}$.