$\square_κ^{<ω}$ holds in $L[{\vec E}]$

良好的尺度与正方形的非紧性

Good Scales and Non-Compactness of Squares

本文研究了正方形原理在奇异基数上的紧性问题，证明了在特定条件下，ℵω上的所有尺度都是良好的，但正方形原理*在ℵω上不成立，同时某些内部可接近性失败。

Subcompact cardinals, squares, and stationary reflection

We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling's hierarchy of weak squares, showing which cases are precluded by…

Local saturation and square everywhere

We show that it is consistent relative to a huge cardinal that for all infinite cardinals $κ$, $\square_κ$ holds and there is a stationary $S \subseteq κ^+$ such that $\mathrm{NS}_{κ^+} \restriction S$ is $κ^{++}$-saturated.

The strong tree property and weak square

We show that it is consistent, relative to $ω$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < ω$ but there are weak square and a very good scale at $\aleph_ω$.

Forcing a $\square(κ)$-like principle to hold at a weakly compact cardinal

Hellsten \cite{MR2026390} proved that when $κ$ is $Π^1_n$-indescribable, the \emph{$n$-club} subsets of $κ$ provide a filter base for the $Π^1_n$-indescribability ideal, and hence can also be used to give a characterization of $Π^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteqκ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteqκ$. By replacing clubs with $n$-clubs in the definition of $\Box(κ)$, one obtains a $\Box(κ)$-like principle $\Box_n(κ)$, a version of which was first considered by Brickhill and Welch …

Regular Ultrapowers at Regular Cardinals

In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{la…