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Tom Benhamou

Publications and source records attributed to Tom Benhamou.

33 records · Page 2Linked to original sources
arXiv2024

On Ultrapowers and Cohesive Ultrafilters

We characterize the Tukey order, the Galvin property/ Cohesive ultrafilters from \cite{Kanamori1978} in terms of ultrapowers. We use this characterization to measure the distance between the Tukey order and other well-known orders of ultrafilters. Secondly, we improve two theorems of Kanamori \cite{Kanamori1978} from the 70's. We then study the point spectrum and the depth spectrum of an ultrafilter, and give a simple positive answer to Kanamori's question \cite[Question 2]{Kanamori1978} starting from a supercompact cardinal. We also prove that a positive answer requires more than $o(κ)=κ^{++}$. Finally, we prove several consistency results regarding the point and depth spectrum.

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arXiv2024

Non-Normal Magidor-Radin Types of Forcings

We develop the non-normal variations of two classical Prikry-type forcings; namely, Magidor and Radin forcings. We generalize the fact that the non-normal Prikry forcing is a projection of the extender-based to a coordinate of the extender to our forcing and the Radin/Magidor-Radin-extender-based forcing from \cite{CarmiMagidorRadin,CarmiRadin}. Then, we show that both the non-normal variation of Magidor and Radin forcings can add a Cohen generic function to every limit point of cofinality $ω$ of the generic club. Second, we show that this phenomenon is limited to the cases where the forcings are not designed to change the cofinality of a measurable $κ$ to $ω_1$. Specifically, in the above-mentioned circumstances these forcings do not project onto any $κ$-distributive forcing. We use that to conclude that the extender-based Radin/Magidor-Radin forcing does not add fresh subsets to $κ$ as well. In the second part of the paper we focus on the natural non-normal variation of Gitik's forcing from \cite[\S3]{GitikNonStationary}. Our main result shows that this poset can be employed to change the cofinality of a measurable cardinal $κ$ to $ω_1$ while introducing a Cohen subset of $κ$.

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arXiv2024

Transferring Compactness

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal $κ$ that is $n$-$d$-stationary for all $n\in ω$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $κ$ being $(n+1)$-$d$-stationary is equivalent to $κ$ being $\mathbfΠ^1_n$-indescribable. We also show that it is consistent that there is a cardinal $κ\leq 2^ω$ such that $P_κ(λ)$ is $n$-stationary for all $λ\geq κ$ and $n\in ω$, answering a question of Sakai.

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arXiv2024

Diamond principles and Tukey-top ultrafilters on a countable set

We provide two types of guessing principles for ultrafilter ($\diamondsuit^{-}_λ(U), \ \diamondsuit^p_λ(U)$) on $ω$ which form subclasses of Tukey-top ultrafilters, and construct such ultrafilters in $ZFC$. These constructions are essentially different from Isbell's construction \cite{Isbell65} of Tukey-top ultrafilters. We prove using the Borel-Cantelli Lemma that full guessing is not possible and rule out several stronger guessing principles e.g. we prove that no Dodd-sound ultrafilters exist on $ω$. We then apply these guessing principles to force a $q$-point which is Tukey-top (answering a question from \cite{Benhanou/Dobrinen23}), and prove that the class of ultrafilters which satisfy $\neg\diamondsuit^{-}_λ$ is closed under Fubini sum. Finally, we show that $\diamondsuit^{-}_λ$ and $\diamondsuit^p_λ$ can be separated.

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arXiv2024

Cofinal types of ultrafilters over measurable cardinals

We develop the theory of cofinal types of ultrafilters over measurable cardinals and establish its connections to Galvin's property. We generalize fundamental results from the countable to the uncountable, but often in surprisingly strengthened forms, and present models with varying structures of the cofinal types of ultrafilters over measurable cardinals.

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arXiv2022

Non-Galvin Filters

We address the question of the consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions \cite[Questions 7.8,7.9]{TomMotiII}, \cite[Question 5]{NegGalSing} and improve theorem \cite[Theorem 2.3]{NegGalSing}.

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arXiv2022

On Cohen and Prikry Forcing Notions

We show that it is possible to add $κ^+-$Cohen subsets to $κ$ with a Prikry forcing over $κ$. This answers a question from \cite{HayutBenhanouGitik}. A strengthening of non-Galvin property is introduced. It is shown to be consistent using a single measurable cardinal which improves a previous result by S. Garti, S. Shelah, and the first author \cite{BenhamouGartieShelah}. A situation with extender-based Prikry forcings is examined. This relates to a question of H. Woodin.

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arXiv2022

Galvin's property at large cardinals and an application to partition calculus

In the first part of this paper, we explore the possibility for a very large cardinal $κ$ to carry a $κ$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $κ$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $κ$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. Finally, we apply this property to obtain consistently new instances of the classical problem in partition calculus $λ\rightarrow(λ,ω+1)^2$.

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arXiv2022

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 $V\subseteq M\subseteq V[G]$.

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arXiv2022

Intermediate Models of Magidor-Radin Forcing- Part II

We continue the work done by the authors and before that by the second author, Kanovei and koepke. We prove that for every set of ordinals $A$ in a Magidor-Radin generic extension using a coherent sequence such that $o^{\vec{U}}(κ)<κ^+$, there is $C'\subseteq C_G$, such that $V[A]=V[C']$. Also we prove that the supremum of a fresh set in a Prikry, tree Prikry, Magidor, Radin-Magidor and Radin forcing, changes cofinality to $ω$.

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arXiv2021

Negating the Galvin Property

We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the effect of Prikry-type forcings on the strong failure of the Galvin property and explores stronger forms of this property in the context of large cardinals

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arXiv2021

The Variety of Projection of a Tree-Prikry Forcing

We study which $κ$-distributive forcing notions of size $κ$ can be embedded into tree Prikry forcing notions with $κ$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $κ$-distributive forcing notion of size $κ$ be extended to a $κ$-complete ultrafilter.

math.LO↗
arXiv2018

Prikry Forcing and Tree Prikry Forcing of Various Filters

In this paper, we answer a question asked in "A minimal Prikry type forcing for singularizing a measurable cardinal" regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.

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