BLC 2010

A new kind of forcing axiom
Mirna Dzamonja
Many advances in set theory have been obtained by using forcing to show that certain statements cannot be true or false in ZFC. The method of forcing is well understood when it comes to statements about omega or omega_1, or on the other hand very large cardinals. At omega_1 it was made particularly useful by indtroduction of forcing axioms, such as Martin Axiom, which make the method accessible to general mathematicians. All this works well for certain kinds of cardinals, but it is known that it cannot be made to work in the same way at some other kinds of cardinals, specifically the successors of singular cardinals. Not even the simplest forcing notions, such as Cohen forcing, make sense at this kind of cardinals. Therefore it is quite challenging to try to have a forcing axiom at such a cardinal. In a joint work with Saharon Shelah we introduce such an axiom, called SUSIFA and apply it in various situations. The talk will present some of these results.