A splitting family at a regular cardinal kappa is defined so that for any subset X of kappa of size kappa there is a member of the family Y which splits X, namely, both X intersect Y and X - Y have size kappa. The splitting number for kappa is the least size of a splitting family at kappa. Large splitting numbers imply an amount of compactness for kappa (from work of Suzuki). Moreover, forcing can increase the splitting number at supercompact kappa (via Zapletal, credited to Kamo).
I will introduce the stationary splitting number, the least size of a stationary splitting family, which is a family of subsets of kappa splitting stationary subsets into sets that are both stationary. This came up during work with Fuchs on what we call Split Principles, and I will give some background on those, and compare the stationary splitting number to the splitting number and what seems to be known.
An exact linear order is one with no non-trivial self-embedding. I shall talk a little bit about these objects and some questions of interest surrounding them, taking us on a path through Ramsey constructions and curious questions in abstract forcing.
Dependent Choice (DC) is one the most useful choice principles with many equivalents (including the Downward Löwenheim–Skolem and the Baire Category Theorem). When we violate the Axiom of Choice via symmetric extensions we often want to preserve at least that much. In this talk we will discuss a few older results about the preservation of DC in generic and symmetric extensions, and we will present a recent breakthrough from a work-in-progress with Jonathan Schilhan.
Dependent Choice (DC) is one the most useful choice principles with many equivalents (including the Downward Löwenheim–Skolem and the Baire Category Theorem). When we violate the Axiom of Choice via symmetric extensions we often want to preserve at least that much. In this talk we will discuss a few older results about the preservation of DC in generic and symmetric extensions, and we will present a recent breakthrough from a work-in-progress with Jonathan Schilhan.
I will present the Kinna-Wagner principles, which are weakenings of the Axiom of Choice and give a proof to what has so far been only a conjecture, which shows how these principles relate to the structure of intermediate models of ZF between a ground model and its forcing extension.
CW complexes are topological spaces built up dimension by dimension from Euclidean cells, with a subset declared to be open if its intersection with each of these cells is open. Unfortunately when you take the product of two CW complexes, the product topology does not in general satisfy this requirement. I will explain when exactly it does; it turns out that it depends on the cardinal $\mathfrak{b}$. For the old hands who have seen this talk multiple times before, there will also be something new, with details that I realised last week I ought to draw out more.
