I will present a higher analogue of the proper forcing axiom, and discuss some of its applications. The higher analogue is an axiom that allows meeting collections of ℵ2 maximal antichains, in specific classes of posets that preserve both ℵ1 and ℵ2.

This talk will include more details and proofs than my talk in the workshop on Forcing Axioms and their Applications. I will quickly survey the previous talk for audience members who were not present in the

previous workshop.