Moti Gitik
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View article: On ultrafilters in ZF models and indecomposable ultrafilters
On ultrafilters in ZF models and indecomposable ultrafilters Open
We use indecomposable ultrafilters to answer some questions from Hayut and Karagila (in Comments Math Univ Carolin 60(2):285–298, 2019). It is shown that the bound on the strength of Usuba (in A note on uniform ultrafilters in choiceless c…
View article: The Galvin property at $κ^{++}$ and not at $κ^+$
The Galvin property at $κ^{++}$ and not at $κ^+$ Open
We construct a $κ-$complete ultrafilter $W$ over $κ$ such that $\neg$Gal$(κ, W, κ^+)$ and Gal$(κ, W, κ^{++})$. This answers a question of T. Benhamou and G. Goldberg.
View article: On countably closed mutually embeddable models
On countably closed mutually embeddable models Open
We show that a measurable cardinal is enough in order to construct two distinct countably closed mutually embeddable models. This answers a question from Eskew, M., et al.: Annals of Pure and Applied Logic, Vol. 175, Issue 1, Part B, 10332…
View article: On ultrafilters in ZF models and indecomposable ultrafilters
On ultrafilters in ZF models and indecomposable ultrafilters Open
We use indecomposable ultrafilters to answer some questions of Hayut, Karagila paper "Spectra of uniformity". It is shown that the bound on the strength by T. Usuba "A note on uniform ultrafilters in choiceless context" is optimal.
View article: Around accumulation points and maximal sequences of indiscernibles
Around accumulation points and maximal sequences of indiscernibles Open
Answering a question of Mitchell (Trans Am Math Soc 329(2):507–530, 1992) we show that a limit of accumulation points can be singular in $${\mathcal {K}}$$ . Some additional constructions are presented.
View article: On fresh sets in iterations of Prikry type forcing notions
On fresh sets in iterations of Prikry type forcing notions Open
We examine the existence (and mostly non-existence) of fresh sets in commonly used iterations of Prikry type forcing notions. Results of [4] are generalized. As an application, a question of a referee of [9] is answered. In addition statio…
View article: The first measurable can be the first inaccessible cardinal
The first measurable can be the first inaccessible cardinal Open
In [8] the second and third authors showed that if the least inaccessible cardinal is the least measurable cardinal, then there is an inner model with $o(κ)\geq2$. In this paper we improve this to $o(κ)\geqκ+1$ and show that if $κ$ is a $κ…
View article: ON EASTON SUPPORT ITERATION OF PRIKRY-TYPE FORCING NOTIONS
ON EASTON SUPPORT ITERATION OF PRIKRY-TYPE FORCING NOTIONS Open
We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extend…
View article: Extender-based Magidor-Radin forcings without top extenders
Extender-based Magidor-Radin forcings without top extenders Open
Continuing \cite{GitJir22}, we develop a version of Extender-based Magidor-Radin forcing where there are no extenders on the top ordinal. As an application, we provide another approach to obtain a failure of SCH on a club subset of an inac…
View article: On Easton support iteration of Prikry type forcing notions
On Easton support iteration of Prikry type forcing notions Open
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 extend…
View article: Non-Galvin Filters
Non-Galvin Filters Open
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 \…
View article: Another method to add a closed unbounded set of former regulars
Another method to add a closed unbounded set of former regulars Open
A club consisting of former regulars is added to an inaccessible cardinal, without changing cofinalities outside it. The initial assumption is optimal. A variation of the Radin forcing without a top measurable cardinal is introduced for th…
View article: On Cohen and Prikry Forcing Notions
On Cohen and Prikry Forcing Notions Open
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 …
View article: ON RESTRICTIONS OF ULTRAFILTERS FROM GENERIC EXTENSIONS TO GROUND MODELS
ON RESTRICTIONS OF ULTRAFILTERS FROM GENERIC EXTENSIONS TO GROUND MODELS Open
Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$ complete ultrafilter 1 , 2 W over $\kappa $ . Set $U=W\cap V$ .
View article: Non-stationary support iterations of Prikry Forcings and Restrictions of Ultrapower Embeddings to the Ground Model
Non-stationary support iterations of Prikry Forcings and Restrictions of Ultrapower Embeddings to the Ground Model Open
We study the nonstationary-support iteration of Prikry forcings below a measurable cardinal κ, characterizing all the normal measures it carries in the generic extension. We then analyze the restriction of ultrapower embeddings, taken with…
View article: The Variety of Projection of a Tree-Prikry Forcing
The Variety of Projection of a Tree-Prikry Forcing Open
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 ope…
View article: Intermediate Models in Magidor-Radin Generic Extensions -- Part I
Intermediate Models in Magidor-Radin Generic Extensions -- Part I Open
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 $o^{\vec{U}}(\kappa)<\kappa$, there is a subset $C'$ of…
View article: Intermediate Models in Magidor-Radin Forcing- Part I
Intermediate Models in Magidor-Radin Forcing- Part I Open
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)
View article: SOME APPLICATIONS OF SUPERCOMPACT EXTENDER BASED FORCINGS TO HOD
SOME APPLICATIONS OF SUPERCOMPACT EXTENDER BASED FORCINGS TO HOD Open
Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V . In particular, a model where all regular uncountable cardinals are measurable in HOD is constructed.
View article: Adding a lot of random reals by adding a few
Adding a lot of random reals by adding a few Open
We study pairs $(V, V_1)$ of models of $ZFC$ such that adding $κ$-many random reals over $V_1$ adds $λ$-many random reals over $V$, for some $λ> κ.$
View article: PCF theory and Woodin cardinals
PCF theory and Woodin cardinals Open
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists. Theorem B: Let kappa be a singul…
View article: On density of old sets in Prikry type extensions
On density of old sets in Prikry type extensions Open
Every set of ordinals of cardinality $\kappa$ in a Prikry extension with a measure over $\kappa$ contains an old set of arbitrarily large cardinality below $\kappa$, and, actually, it can be split into countably many old sets. What about s…
View article: ON THE SPLITTING NUMBER AT REGULAR CARDINALS
ON THE SPLITTING NUMBER AT REGULAR CARDINALS Open
Let κ , λ be regular uncountable cardinals such that λ > κ + is not a successor of a singular cardinal of low cofinality. We construct a generic extension with s ( κ ) = λ starting from a ground model in which o ( κ ) = λ and prove that as…
View article: Adding a lot of Cohen reals by adding a few II
Adding a lot of Cohen reals by adding a few II Open
We study pairs $(V, V_{1})$, $V \subseteq V_1$, of models of $ZFC$ such that adding $κ-$many Cohen reals over $V_{1}$ adds $λ-$many Cohen reals over $V$ for some $λ> κ$.