arXiv subjects
Publications and source records attributed to Moti Gitik.
We construct a $\kappa-$complete ultrafilter $W$ over $\kappa$ such that $\neg$Gal$(\kappa, W, \kappa^+)$ and Gal$(\kappa, W, \kappa^{++})$. This answers a question of T. Benhamou and G. Goldberg.
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.
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 stationary sets preservation is addressed.
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 $o(\kappa)\geq2$. In this paper we improve this to $o(\kappa)\geq\kappa+1$ and show that if $\kappa$ is a $\kappa^{++}$-supercompact cardinal, then there is a symmetric extension in which it is the least inaccessible and the least measurable cardinal.
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 inaccessible cardinal, and a model where the cardinal arithmetic behaviors are different on stationary classes, whose union is the club, is provided. The cardinals and the cofinalities outside the clubs are not affected by the forcings.
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.
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}.
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 this.
We show that it is possible to add $\kappa^+-$Cohen subsets to $\kappa$ with a Prikry forcing over $\kappa$. 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.
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.
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 such a normal measure in the generic extension, to the ground model. We prove that every such restriction is an iterated ultrapwer of the ground model, and provide a sufficient condition for its definability there. This is done without core-model theoretic arguments: the assumption that GCH holds in the ground model up to κsuffices.
We continue Gitik, Kanovei and Koepke's work and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.
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 $o^{\vec{U}}(\kappa)<\kappa^+$, 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 $\omega$.
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 the Magidor club such that $V[A]=V[C']$. Also we classify all intermediate $ZFC$ transitive models $V\subseteq M\subseteq V[G]$.
We prove the consistency of the statement $\mathfrak{u}_{\aleph_ω}<2^{\aleph_ω}$. We show that the consistency strength of this statement is exactly a measurable cardinal $μ$ so that $o(μ)=μ^{++}$.
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 $λ> κ.$
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.
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 assuming $\neg 0^¶$, $s(κ) = λ$ implies that $o(κ) \geq λ$ in the core model.