We will discuss Viale-Weiss paper ``On the consistency strength of the proper forcing axiom".

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We give some properties of the \kappa-topologies over 2^{\lambda} introduced earlier (for \kappa \leq \lamba), define a Cantor-Bendixon process for forests, and begin work on showing the consistency of Cantor-Bendixon theorem analogues for closed subsets of 2^{\kappa} and P_{\kappa^+}\lambda, for \kappa regular.

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. This week we finish the proof of the fact that \kappa-closed forcings don't add branches to \kappa-scattered subsets of 2^{\kappa}. We then introduce a collection of topologies over 2^{\lambda} whose restrictions to P_{\kappa}\lambda have some desirable properties. These topologies will rely on the notion of a P_{\kappa}\lambda-forest, which is a natural generalization of a tree. 

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We focus in particular this week on constructing certain types of trees in 2^{<\kappa} for uncountable \kappa which exhibit fundamentally different behavior than trees in 2^{<\omega} can, from the perspective of adding branches, cardinal dichotomies, etc. We also generalize the games previously discussed, and introduce alternative notions of \kappa-perfect and \kappa-scattered.

We will initially discuss games played on subsets of the Cantor space, for which the existence or nonexistence of winning strategies for certain players can provide a characterization of perfectness or scatteredness. We will also give an old characterization of the type of trees in 2^{<\omega} through which outer models can add branches. Finally, we will make some observations about the nature of some generalizations of these topics to the 2^{\kappa} spaces.