Towards combining “compactness” and “hugeness” properties at ω2, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at ω2 using Neeman's forcing. On the other hand, we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating such forcing.

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.

Abstract 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 ${\mathbb R}$ .

Abstract From large cardinals we show the consistency of normal, fine, κ-complete λ-dense ideals on ${{\cal P}_\kappa }\left( \lambda \right)$ for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.