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

Small forcings and Cohen reals

We prove it consistent relative to ZFC that all nontrivial forcings of size $\aleph _1$ add a Cohen real

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

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

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]

A New Proof of Kunen's Inconsistency

Using elementary pcf, we show that there is no $j:V\to M,$ $M$ transitive, $j\lambda =\lambda >crit(j),$ $j^{\prime \prime}\lambda \in M.

Splitting number and the core model

We can generalize the definition of {\it splitting number } $s(\kappa )$ for $\kappa$ uncountable regular: s(\kappa )=min\{ |\Cal S|:\Cal S\subset \Cal P(\kappa ) \forall a\in \kappa ^\kappa \exists b\in \Cal S |a\cap b|=|a\setminus b|=\kappa\} However,$\exists \kappa>\aleph_0$ $s(\kappa )>\kappa ^+$ becomes a considerable hypothesis,shown consistent from a supercompact.We show that it implies inner models of $\exists \alpha :o(\alpha )=\alpha ^{++}

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

More on the cut and choose game

We improve some ancient results of Velickovic on the cut and choose (c&c) game on complete Boolean algebras. (1) If Nonempty has a winning strategy for c&c game on $B$ then $B$ is semiproper. (2) If Nonempty has a winning strategy and $B$ has $2^{\aleph _0}$ -c.c. then Nonempty has a winning strategy in the descending chain game. (3) Cons ($B$ is $\aleph _1$-distributive implies Nonempty has a winning strategy in c&c on $B$ ) We also give some new examples of forcings where Nonempty has or does not have a winning strategy in c&c game