Parametrized Measuring and Club Guessing

On squares, outside guessing of clubs and I_{<f}[lambda]

Suppose that lambda = mu^+. We consider two aspects of the square property on subsets of lambda. First, we have results which show e.g. that for aleph_0 <= kappa =cf (kappa)< mu, the equality cf([mu]^{<= kappa}, subseteq)= mu is a sufficient condition for the set of elements of lambda whose cofinality is bounded by kappa, to be split into the union of mu sets with squares. Secondly, we introduce a certain weak version of the square property and prove that if mu is a strong limit, then this weak square property holds on lambda without any additional assumptions

On versions of clubsuit on cardinals larger than aleph_1

We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in math.LO/9808140 shows that if 2^lambda=lambd…

Measuring club-sequences with a large continuum

One of the most frustrating problems faced by set theorists working with iterated proper forcing is the lack of techniques for producing models in which the continuum has size greater than the second uncountable cardinal. In this paper we solve this problem in the specific case of measuring, a very strong negation of Club Guessing introduced by Justin Moore.

On club-like principles on regular cardinals above $\beth_ω$

We prove that for regular $λ$ above a strong limit singular $μ$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $λ$ and $μ$ there are coboundedly many regular $κ<μ$ such that $\clubsuit^-(S^λ_κ)$ holds whenever $λ=λ^{<κ}$.

来自$\clubsuit$型原理的非平凡导出极限

Nonvanishing derived limits from $\clubsuit$-type principles

本研究聚焦组合集合论中阿贝尔群逆系统的非平凡导出极限问题，改用$\clubsuit$型原理及其弱化形式开展探究，突破了以往依赖弱钻石类猜测原理的研究路径，拓展了相关非平凡性结果的推导工具范畴。

A guessing principle from a Souslin tree, with applications to topology

We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces. Our main result states that strong instances of $\clubsuit_{AD}$ follow from the existence of a Souslin tree. It is also shown that the weakest instance of $\clubsuit_{AD}$ does not follow from the existence of an almost Souslin tree. As an application, we obtain a simple, de Caux type proof of Rudin's result…

猜测泛化性——从不同视角审视参数化钻石原则

Guessing genericity -- looking at parametrized diamonds from a different perspective

本文从模型论与力迫论视角提出一族参数化钻石公理，揭示不同原则间的深层联系，并证明测度代数力迫保持$\diamondsuit(\mathfrak{d})$变体，改进已有结果。

A continuation of [DjSh:691]

We continue to investigate club guessing (see [Sh:g, Ch III] and [Sh:e, Ch VI]), continuing Dzamonja and Shelah [DjSh:691]

Specializing Trees with Small Approximations I

Assuming $\rm PFA$, we shall use internally club $ω_1$-guessing models as side conditions to show that for every tree $T$ of height $ω_2$ without cofinal branches, there is a proper and $\aleph_2$-preserving forcing notion with finite conditions which specialises $T$. Moreover, the forcing has the $ω_1$-approximation property.

Parametrized $\diamondsuit$ principles

We will present a collection of guessing principles which have a similar relationship to $\diamond$ as cardinal invariants of the continuum have to $\CH$. The purpose is to provide a means for systematically analyzing $\diamond$ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of $\CH$ and $\diamond$ in models such as those of Laver, Miller, and Sacks.

一个参数化的$\diamondsuit$用于Laver性质和$\mathcal P(ω)/\mathrm{Fin}$的非平凡自同构

A parametrized $\diamondsuit$ for the Laver property and nontrivial automorphisms of $\mathcal P(ω)/\mathrm{Fin}$

本文提出参数化$\diamondsuit(\mathsf{LP})$原理，证明其在特定forcing模型中成立，并展示其导致$\mathcal P(ω)/\mathrm{Fin}$存在非平凡自同构。