Abstract We show that it is independent whether club $\kappa $ -Miller forcing preserves $\kappa ^{++}$ . We show that under $\kappa ^{<\kappa }> \kappa $ , club $\kappa $ -Miller forcing collapses $\kappa ^{<\kappa }$ to $\kappa $ . Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter $\kappa $ -Miller forcing does not have the Laver property.

Abstract We give an affirmative answer to Brendle's and Hrušák's question of whether the club principle together with is consistent. We work with a class of axiomAforcings with countable conditions such that is determined by finitely many elements in the conditionspandqand that all strengthenings of a condition are subsets, and replace many names by actual sets. There are two types of technique: one for tree-like forcings and one for forcings with creatures that are translated into trees. Both lead to new models of the club principle.

Abstract We prove the statement in the title.

Abstract We specialise Aronszajn trees by an ωω-bounding forcing that adds reals. We work with creature forcings on uncountable spaces. As an application of these notions of forcing, we answer a question of Moore, Hrušák and Džamonja whether implies the existence of a Souslin tree in a negative way by showing that “ and every Aronszajn tree is special” is consistent relative to ZFC.

Abstract We show that in the models ofu< ∂ from [14] there are infinitely many near-coherence classes of ultrafilters, thus answering Banakh's and Blass' Question 30 of [3] negatively. By an unpublished result of Canjar, there are at least two classes in these models.

Abstract We prove the consistency result from the title. By forcing we construct a model of g = ℵ1, b = cf(Sym(ω)) = ℵ2.

Abstract We prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve three cardinal characteristics of the continuum, the splitting numbers, the unsplitting numberr, and the groupwise density numberg. We also prove some related results for reduced powers with respect to filters other than ultrafilters.

Abstract We show: The procedure mentioned in the title is often impossible. It requires at least an inner model with a measurable cardinal. The consistency strength of changing and from a regularκto some regularδ<κis a measurable of Mitchell orderδ. There is an application to Cichoń's diagram.

Abstract There are inequalities between cardinal characteristics of the continuum that are true in any model of ZFC, but without a Borel morphism proving the inequality. We answer some questions from Blass [1].