Starting from a supercompact, we construct a model in which SCH fails at $\aleph_\omega$ and there is a bad scale at $\aleph_\omega$. The existence of a bad scale implies the failure of weak square. The construction uses two Prikry type forcings defined in different ground models and a suitably defined projection between them. This is joint work with Spencer Unger.

It is hard to find analogues of MA in which aleph_1 is replaced by the successor of a singular cardinal because
a) The consequences of MA-like axioms have large consistency strength
b) There is no satisfactory analogue of finite support ccc iteration

Dzamonja and Shelah found an ingenious approach to proving results of this general kind. I will outline their work and then describe some recent joint work with Dzamonja and Morgan, aimed at bringing results of this kind down to aleph_{omega+1}

Basic theory of modified Prikry forcing will be presented. In particular, the proof of Prikry lemma for modified Prikry forcing will be discussed.

Given a regular cardinal $\kappa$, an uncountably cofinal ordinal $\nu<\kappa$ is a reflection point of the stationry set $S\subseteq\kappa$ just in the case where $S\cap\alpha$ is stationary in $\alpha$. Starting from ininitely many supercompact cardinals, Magidor constructed a model of set theory where every stationary $S\subseteq\aleph_{\omega+1}$ has a reflection point. In this series of talks we present a construction of a model of set theory where we obtain a large amount of stationary reflection (although not full) using a significantly weaker large cardinal hypothesis. We start from a quasicompact (quasicompactness is a large cardinal hypothesis significantly weaker than any nontrivial variant of supercompactness) cardinal $\kappa$ and use modified Prikry forcing to turn $\kappa$ into $\aleph_{\omega+1}$. We then show that in the resulting model every stationray $S\subeteq\aleph_{\omega+1}$ not concentrating on ordinals of ground model cofinality $\kappa$ has a reflection point.

Given a measurable cardinal \kappa, any \Sigma^1_3 statement is absolute for generic extensions via forcings of size <\kappa. A proof of this classical theorem will be presented.