Extender Based Radin Forcing

Intermediate Models in Magidor-Radin Forcing- Part I

We continue the work done by Gitik, Kanovei, Koepke, and later by the authors. We prove that for every set $A$ in a Magidor-Radin generic extension using a coherent sequence such that $o^{\vec{U}}(κ)<κ$, there is a subset $C'$ of the Magidor club such that $V[A]=V[C']$. Also we classify all intermediate $ZFC$ transitive models $V\subseteq M\subseteq V[G]$.

Some applications of Supercompact Extender Based Forcings to HOD

Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.

Forcing with overlapping supercompact extenders

We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.

Compactness and Guessing Principles in the Radin Extensions

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $κ$, if $κ$ is weakly compact, then $\diamondsuit(κ)$ holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently …

在非重叠扩展器模型上用力迫保持超幂公理

Preserving the Ultrapower Axiom by Forcing over Nonoverlapping Extender Models

本文通过构造离散乘积力迫，在非重叠扩展器模型上保持超幂公理至强基数层次，并证明其与特定V≠HOD_X条件一致，解答了Goldberg的开放问题。

Prikry On Extenders, Revisited

We present a modification to the Prikry on Extenders forcing notion allowing the blow up of the power set of a large cardinal, change its cofinality to omega without adding bounded subsets, working directly from arbitrary extender (e.g., n-huge extender). Using this forcing, starting from a superstrong cardinal, we construct a model in which the added Prikry sequences are a scale in the normal Prikry sequence.

Prikry型力迫与Bukovský-Dehornoy现象

Prikry type forcings and the Bukovský-Dehornoy phenomena

本文系统梳理了Bukovský和Dehornoy独立发现的常见力迫现象，先给出Prikry型力迫的基本结果，再将其推广到Gitik-Sharon力迫、基于Extender的Prikry力迫、带交错坍缩的Prikry力迫及Radin力迫。

Diamonds, Compactness, and Measure Sequences

We establish the consistency of the failure of the diamond principle on a cardinal $κ$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of $κ$ in a Radin generic extension.

On Singular Stationarity II (tight stationarity and extenders-based methods)

We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in \cite{ForMag-MS}. We obtain two consistency results which show that it is possible for a sequence of regular cardinals $( κ_n )_{n < ω}$ to have the property that for every sequence $\vec{S}$, of some fixed-cofinality stationary sets $S_n \subseteq κ_n$, $\vec{S}$ is tightly stationary in a generic extension. The results are obtained using variations of the short-extenders forcing method.

On Easton support iteration of Prikry type forcing notions

We consider here Easton support iterations of Prikry type forcing notions. New ways of constructing normal ultrafilters in extensions are presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.

在每一个奇异基数上都存在一个小的超滤子数

A Small Ultrafilter Number at Every Singular Cardinal

本文通过重叠强扩展子forcing方法，构造了一个模型，其中最小的inaccessible基数$κ$满足GCH在正则数上，SCH在奇异数上失败，并且所有奇异数的超滤子数都小。

论 Cohen 与 Prikry 力迫概念

On Cohen and Prikry Forcing Notions

本文证明用 κ 上的 Prikry 力迫可添加 κ⁺ 个 Cohen 子集，并引入非 Galvin 性质的强化，仅需单个可测基数即证明其相容性，同时讨论基于 extender 的 Prikry 力迫。