Splitting stationary sets from weak forms of Choice

Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below $\lambda$ of cofinality $\theta$ into $\lambda$ many stationary sets, where $\theta < \lambda$ are regular cardinals. This is a continuation of \cite{Sh835}

A framework for forcing constructions at successors of singular cardinals

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of uncountable cofinality, while κ^+ enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ of uncountable cofinality where SCH fails and such that there is a collection of size less than 2^{κ^+} of graphs on κ^+ such that any graph on κ^+ embeds into one of the graphs in the collection

Potential isomorphism and semi-proper trees

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notions of semi-proper and weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We prove the existence of semi-proper trees under certain cardinal arithmetic assumptions. We also show the consistency of the non-existence of weakly semi-proper trees assuming the consistency of some large cardinals

Embeddings into outer models

We explore the possibilities for elementary embeddings $j : M \to N$, where $M$ and $N$ are models of ZFC with the same ordinals, $M \subseteq N$, and $N$ has access to large pieces of $j$. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line $\mathbb R$