Rahman Mohammadpour
In this short note, we shall prove some observations regarding the connection between indestructible $ω_1$-guessing models and the $ω_1$-approximation property of forcing notions.
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继续探索
We introduce a stronger version of an $ω_1$-guessing model, which we call an indestructibly $ω_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly $ω_1$-guessing models. This principle, which follows from PFA, captures many of the consequences of PFA, including the Suslin hypothesis and the singular cardinal hypothesis. We prove that IGMP is consistent with the continuum being arbitrarily large.
On Indestructible Strongly Guessing Models
本文提出了GM+原则的强化版本SGM+，并在超紧基数假设下证明了其与ZFC的一致性。该原则连同温和的基数算术假设表明，任何增加ω₂新子集的力迫必然增加实数或崩塌基数，部分回答了Abraham问题并扩展了Todorčević的结果。
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is a…
Almost Kurepa Suslin trees and destructibility of the Guessing Model Property
本文证明了在ω₂处猜测模型原理与几乎Kurepa Suslin树存在的一致性，并展示了该原理可能被cccforcing破坏，同时探讨了弱Kurepa树与树性质的关系。
Strong tree properties, Kurepa trees, and guessing models
本文研究广义树性质与猜测模型性质，提出几乎猜测性质以刻画纤细树性质，并证明其推论、在 Mitchell 模型中的成立性及相关一致性结果。
A concept of "guessability" is defined for sets of sequences of naturals. Eventually, these sets are thoroughly characterized. To do this, a nonstandard logic is developed, a logic containing symbols for the ellipsis as well as for functions without fixed arity. New proofs are given for some seemingly-unrelated known results.
In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $κ\ge ω_2$, $\textsf{ISP}(κ)$ implies that $\textsf{SCH}$ holds above $κ$, and (3) forcing posets which have the $ω_1$-approximation property also have the countable covering property. These results solve open problems of Viale and Hachtman-Sinapova.
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
A Road To Compactness Through Guessing Models
本文综述猜测模型如何提供紧致性现象以解决ZFC未决数学命题，并对其结果进行推广、扩展与应用探讨。
We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $ω_1$-approximation property. We prove that the existence of stationarily many $ω_1$-guessing models in $P_{ω_2}(H(θ))$, for sufficiently large cardinals $θ$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.
There are several examples in the literature showing that compactness-like properties of a cardinal $κ$ cause poor behavior of some generic ultrapowers which have critical point $κ$ (Burke \cite{MR1472122} when $κ$ is a supercompact cardinal; Foreman-Magidor \cite{MR1359154} when $κ= ω_2$ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\vec{\mathcal{I}}$ is a tower of ideals which concentrates on the class $GIC_{ω_1}$ of $ω_1$-guessing, internally club sets, then $\vec{\mathcal{I}}$ is not presaturated…
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.
Specialising Trees With Small Approximations II
本文证明正则基数上的理想带来无共尾分支树的可特化紧致性，并借助 Neeman 边条件方法建立其与大量猜测模型的相容性，从而将既有特化定理推广到高度 $κ^{++}$ 的无分支树。
Classical Namba forcing can have the weak countable approximation property
该研究从不可达基数出发证明经典Namba力迫可具有弱$\omega_1$-逼近性质，推导了$\mathsf{MM}$下相关猜测模型的存在性，解决了Cox、Krueger及Adolf等人的多个集合论力迫相关问题。
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