Fri Feb 15, 2013
60 Evans Hall, 4:10–6 PM
Itay Neeman (UCLA)
Forcing axioms are statements about the existence of filters over partially ordered sets, meeting given collections of dense open subsets. They have been used for several decades as center points for consistency proofs. A forcing axiom is shown consistent, typically through an iterated forcing construction, and then the consistency of other statements can be established by deriving them from the axiom.
In my talk I will begin with background on forcing and on some of the most important forcing axioms, including Martin’s Axiom (MA), and the Proper Forcing Axiom (PFA). I will discuss some key points in the axioms’ consistency proofs, and difficulties in adapting the proofs to obtain analogues of PFA that involve meeting more than ℵ1 dense open sets. I will end with recent work on new consistency proofs, and higher analogues of PFA.