Gunter Fuchs, Joel David Hamkins
We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.
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Examples for Souslin forcing
我们给出了一个模型，其中有一个ccc Souslin强迫不满足Knaster条件。接下来，我们提出了一个模型，其中有一个sigma连接的不是sigma中心的Souslin强迫，使其所有的小子集都是sigma中心的，但马丁公理失败了这个顺序。此外，我们构造了一个完全非齐次的Souslin强迫，我们建立了一个Souslin强迫，这是正确的，但不是ccc，不包含一个完满集的相互不相容的条件。最后，我们证明了强迫的σ ^[1_2]-概念可能不是牢不可破的ccc。
An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion --- Cohen forcing --- adds an $\aleph_1$-Souslin tree. In this paper, we identify a rather large class of notions of forcing that, assuming a GCH-type assumption, add a $λ^+$-Souslin tree. This class includes Prikry, Magidor and Radin forcing.
We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
We describe a method of building ``nice'' sigma-ideals from Souslin ccc forcing notions. [These notes were written down in 1992, but were not submitted to any journal. In a slightly modified form, they were incorporated to: T. Bartoszynski and H. Judah, Set Theory: on the structure of the real line, A K Peters, Wellesley, MA, 1995; pages 193-203.]
We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.
It is consistent that there exists a Souslin tree $T$ such that after forcing with it, $T$ becomes an almost Souslin Kurepa tree. This answers a question of Zakrzewski.
How special are Cohen and random forcings i.e. Boolean algebras of the family of subsets of reals modulo meagre or null
认为这两个力迫概念--科恩和随机--（相当于相应的布尔代数波莱尔（R）/（贫集）、波莱尔（R）/（空集））是特殊的感觉，可能是古老而广泛的。一个合理的解释是显示他们独特的，或“最小的”或至少在一个家庭的“好强迫”像博雷尔的特点。我们将解释为Souslin“好”建议犹大Shelah [JdSh 292]。我们把Souslin强迫族分成两部分，并期望：在第一部分中，即那些添加一些非支配实数的部分中，Cohen是最小的（=低于每一个），而在其余部分中，随机是非常典型的，甚至是唯一的。关于第二类，我们有弱的结果，关于第一类，我们的结果看起来令人满意。我们有两个主要的结果：一个（1.14）说科恩强迫在第一类中是“最小的”，另一个（1.10）说所有的c.c.c. Souslin强迫具有Cohen强迫和Random实强迫共有的性质，因此它对随机强迫的特殊性问题给出了一个弱的答案，但对所有的c.c.c.苏斯林强迫。
On the rigidity of Souslin trees and their generic branches
该数理逻辑研究证明了存在满足特定刚性与Kurepa性质的Souslin树的相容性，解答了Fuchs的相关问题，还证明了与菱形原理相容的Souslin树稠密子集性质，回应了Baumgartner的相关问题。
More notions of forcing add a square
研究在连续统假设下添加$\square_{\aleph_1}$的力迫概念$\mathbb P$，证明其可实现为$\aleph_2$-苏斯林树，更一般地，$\square_\lambda$可由$\lambda^+$-苏斯林树添加，构造统一且可扩展到不可达基数。
We analyze the forcing notion $\mathcal P$ of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form $H_θ$. We show that forcing with this poset adds a Kurepa tree $T$. Moreover, if $\mathcal P_c$ is a suborder of $\mathcal P$ containing only continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level set of any antichain in $T$ is not stationary in $ω_1$.
Boban Velickovic asked the following question: Is there a nontrivial forcing notion with the Sacks property which is also ccc? A ``definable'' variant of this question has been answered in [Sh:480] (math.LO/9303208): Every nontrivial Souslin forcing notion which has the Sacks property has an uncountable antichain. Here we show that it is consistent that every nontrivial forcing notion which has the Sacks property has an uncountable antichain. Independently, Velickovic has also proved the consistency of this statement.
Stationary tower forcings and universally Baire sets
本文研究平稳塔力迫，通过给出平稳塔所需条件，建立 $L(\mathbb{R})$ 中实数集具有普遍 Baire 性的判定依据。
We show if we use countable support iteration of forcing notions not adding reals that satisfy additional conditions, then the limit forcing does not add reals. As a result we prove that we can amalgamate two earlier methods and prove the consistency with ZFC + GCH of two statements gotten separately earlier: Souslin hypothesis and non-club guessing. We also answer a question of Justin Moore by proving the consistency of one further case of "strong failure of club guessing" with GCH.
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…
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group and so on, iterating transfinitely. Each group maps into the next using inner automorphisms and one takes a direct limit at limit stages. The question is whether the process ever terminates in a fixed point, a group which is isomorphic to its automorphism group by the natural map. In this talk I will prove that every group has a terminating automorphism tower. After this, I will discuss the set-theoretic aspects of the height of the automorphism tower of a group, and sketch m…
A new model for all $C$-sequences are trivial
本文构造了一个新模型，使所有 $C$-序列均为平凡，同时存在具有完全消失层的 $κ$-Souslin 树，从而回答了 Lambie-Hanson 和 Rinot 的问题，并实现了紧致性与非紧致性的最优组合。
If $G$ is a centreless group, then $τ(G)$ denotes the height of the automorphism tower of $G$. We prove that it is consistent that for every cardinal $λ$ and every ordinal $α< λ$, there exists a centreless group $G$ such that (a) $τ(G) = α$; and (b) if $β$ is any ordinal such that $1 \leq β< λ$, then there exists a notion of forcing $P$, which preserves cofinalities and cardinalities, such that $τ(G) = β$ in the corresponding generic extension $V^{P}$.
We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. The main results is that the forcing axiom MM^{++} (also known as MM^{+ω_1}) decides the Π_2-theory of H_{ω_2} with respect to stationary set preserving forcings. We argue that this is a close to optimal generalization to H_{ω_2} of Woodin's absoluteness results for L(R).
We prove that the theory of the models constructible using finitely many cofinality quantifiers - $C_{λ_{1},...,λ_{n}}^{*}$ and $C_{<λ_{1},...,<λ_{n}}^{*}$ for $λ_{1},...,λ_{n}$ regular cardinals - is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to $<μ$-closed sets.
A note on adding isomorphisms and the pseudointersection number
本文证明对任意塔存在稠密集，使任何合理的同构添加力迫都会为该塔添加伪交，从而表明特定公理在已知模型中成立，为著名问题提供了新见解。
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