This paper investigates when it is possible for a partial ordering P to force Pκ(λ) \ V to be stationary in VP. It follows from a result of Gitik that whenever P adds a new real, then Pκ(λ) \ V is stationary in VP for each regular uncountable cardinal κ in VP and all cardinals λ > κ in VP [4]. However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The (...) following is equiconsistent with a proper class of ω₁-Erdős cardinals: If P is N₁-Cohen forcing, then Pκ(λ) \ V is stationary in VP, for all regular κ ≥ N₂ and all λ > κ. The following is equiconsistent with an ω₁-Erdős cardinal: If P is N₁-Cohen forcing, then PN₂ (N₃) \ V is stationary in VP. The following is equiconsistent with κ measurable cardinals: If P is κ-Cohen forcing, then Pκ + (Nκ \ V is stationary in VP. (shrink)

If a˜cardinal κ1, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ0 and its new cofinality, ρ, is less than κ0, then, under some additional assumptions, each cardinal λ>κ1 less than cc/[κ1]<κ1) is collapsed to κ0 as well. If in addition N=M[f], where f : ρ→κ1 is an unbounded mapping, then N is a˜λ=κ0-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba.

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse $$2^\omega $$ 2 ω to $$\omega $$ ω, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

This paper attempts to present and organize several problems in the theory of Singular Cardinals. The most famous problems in the area (bounds for the ℶ-function at singular cardinals) are well known to all mathematicians with even a rudimentary interest in set theory. However, it is less well known that the combinatorics of singular cardinals is a thriving area with results and problems that do not depend on a solution of the Singular Cardinals Hypothesis. We present here an annotated collection (...) of representative problems with some references. Where the problems are novel, attribution is attempted and it is noted where money is attached to particular problems. Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF theory. Important subthemes were mutual stationarity, Aronszajn trees, and superatomic Boolean Algebras. The author notes considerable overlap between this paper and the unpublished report submitted to the Banff Center for the Workshop on Singular Cardinals Combinatorics, May 1–5, 2004. (shrink)