Higher-dimensional forcing

Methods in Higher Forcing Axioms (Workshop Notes)

Methods of Higher Forcing Axioms was a small workshop in Norwich, taking place between 10--12 of September, 2019. The goal was to encourage future collaborations, and create more focused threads of research on the topic of higher forcing axioms. This is an improved version of the notes taken during the meeting by Asaf Karagila.

Forcing Diamond 及其在可迭代性中的应用

Forcing Diamond and Applications to Iterability

本文证明高阶Sacks力迫与club Miller力迫可添加diamond序列，并肯定回答了κ-Sacks力迫的κ-支撑迭代不坍缩κ+且为κ-proper的长期问题，结果适用于其他高阶树力迫。

Higher random indestructibility of MAD families

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $κ$ is indestructible by the higher random forcing $\mathbb Q_κ$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_κ) = \mathfrak b_κ= \mathfrak c_κ$ implies the existence $\mathbb Q_κ$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.

Forcings constructed along morasses

In a previous paper, we introduced a way of constructing a forcing along a simplified gap-1 morass such that the forcing satisfies a chain condition. Now, we generalize this to gap-2 morasses. As an application, we prove that GCH is consistent with the existence of a 0-dimensional Hausdorff topology on $ω_3$ which has spread $ω_1$.

Are all natural numbers the same

This is a report on state-of-the-art on the question of developing higher analogues of the forcing axiom PFA. Recently there have been several attempts to develop forcing axioms analogous to the proper forcing axiom (PFA) for cardinals of the form aleph_n where n > 1. We investigate the difficulties of doing this and survey some of the successes

Higher-dimensional Delta-systems

We investigate higher-dimensional $Δ$-systems indexed by finite sets of ordinals, isolating a particular definition thereof and proving a higher-dimensional version of the classical $Δ$-system lemma. We focus in particular on systems that consist of sets of ordinals, in which case useful order-theoretic uniformities can be ensured. We then present three applications of these higher-dimensional $Δ$-systems to problems involving the interplay between forcing and partition relations on the reals.