An interpolation theorem

Whenever x is a tame cardinal invariant and ZFC+large cardinals proves that x=aleph one implies WCG then ZFC+large cardinals proves that x=aleph one implies b=aleph one, and b=aleph one implies WCG. Here WCG is a certain prediction principle on omega one. This theorem is one of the many possible interpolation theorems of this kind

Proper forcing and rectangular Ramsey theorems

I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.Comment: 16 page

Hypergraphs and proper forcing

Given a Polish space X and a countable family of analytic hypergraphs on X, I consider the sigma-ideal generated by Borel sets which are anticliques in at least one hypergraph in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies. For this broad class of posets, most fusion arguments and iteration preservation arguments can be replaced with simple combinatorial considerations concerning the hypergraphs

Countable Support Iteration Revisited

Whenever P is a proper definable forcing for adding a real, the countable support iteration of P has all the preservation properties it can possibly have, within a wide syntactically identified class of properties

Duality Chipped

Whenever I is a projectively generated projectively defined sigma ideal on the reals, if ZFC+large cardinals proves cov(I)=continuum then ZFC+large cardinals proves non(I)<aleph four

Forcing with ideals of closed sets

Let I be a sigma-ideal sigma-generated by a projective collection of closed sets. The forcing with I-positive Borel sets is proper and adds a single real r of an almost minimal degree: if s is a real in V[r] then s is Cohen generic over V or V[s]=V[r]

Potential theory and forcing

We isolate a combinatorial property of capacities leading to a construction of proper forcings. Then we show that many classical capacities such as the Newtonian capacity satisfy the property.Comment: 21 page

Bounded Namba forcing axiom may fail

In a sigma-closed forcing extension, the bounded forcing axiom for Namba forcing fails. This answers a question of Justin Tatch Moore

Interpreter fr topologists

Let M be a transitive model of set theory. There is a canonical interpretation functor between the category of regular Hausdorff, continuous open images of Cech-complete spaces of M and the same category in V, preserving many concepts of topology, functional analysis, and dynamics. The functor can be further canonically extended to the category of Borel subspaces. This greatly simplifies and extends similar results of Fremlin

Two preservation theorems

I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.Comment: 21 page