Related papers: Dowker filters and Magidor forcing

We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…

We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e., the cofinality of ^{lambda}lambda, is strictly bigger than cov_lambda(meagre), i.e. the minimal number of nowhere dense subsets of…

Filter stability is a classical problem in the study of partially observed Markov processes (POMP), also known as hidden Markov models (HMM). For a POMP, an incorrectly initialized non-linear filter is said to be (asymptotically) stable if…

We look at various questions related to filtrations in $p$-adic Hodgetheory, using a blend of building and Tannakian tools. Specifically,Fontaine and Rapoport used a theorem of Laffaille on filtered isocrystalsto establish a converse of…

Answering a question of Sakai, we show that the existence of an $\omega_1$-Erd\H{o}s cardinal suffices to obtain the consistency of Chang's Conjecture with $\square_{\omega_1, 2}$. By a result of Donder this is best possible. We also give…

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…

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $\kappa$, updating some classical constructions in the process. This includes models of…

Using a theorem from pcf theory, we show that for any singular cardinal nu, the product of the Cohen forcing notions on kappa, kappa < nu adds a generic for the Cohen forcing notion on nu^+. This solves Problem 5.1 in Miller's list…

We continue [Sh:b, Ch XIII] and [Sh:410]. Let W be an inner model of ZFC. Let kappa be a cardinal in V. We say that kappa-covering holds between V and W iff for all X in V with X subseteq ON and V models |X|< kappa, there exists Y in W such…

Under the assumption that $\delta$ is a Woodin cardinal and $\GCH$ holds, I show that if $F$ is any class function from the regular cardinals to the cardinals such that (1) $\kappa<\cf(F(\kappa))$, (2) $\kappa<\lambda$ implies…

We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(\kappa^+,\kappa)\notcc(\aleph\_1,\aleph\_0)$ when $\kappa$ is supercompact. The…

We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (\partial^3_x u +u\partial_x u +\lambda u)dt = f dt+\Phi dW_t$ on ${\mathbb R}$ where $f$ is a deterministic force. We prove that the Feller…

We succeed to say something on the identities of (mu^+, mu) when mu>theta>cf(mu), mu strong limit theta--compact. This hopefully will help to prove the consistency of ``some pair (mu^+,mu) is not compact'', however, this has not been…

This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $\kappa$. In particular, using a concept called Covering Matrices, we show…

We show that it is equiconsistent with $\mathsf{ZF}$ that Fodor's lemma fails everywhere, and furthermore that the club filter on every regular cardinal is not even $\sigma$-complete. Moreover, these failures can be controlled in a very…

Given sets $X,Y$ and a regular cardinal $\mu$, let $\Phi(X,Y,\mu)$ be the statement that for any function $f : X \times Y \to \mu$, there are functions $g_1 : X \to \mu$ and $g_2 : Y \to \mu$ such that or all $(x,y) \in X \times Y$,…

A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed…

Let $\kappa$ be an infinite cardinal. Then, forcing with $\mathbb{R}(\kappa)$$\times$$\mathbb{R}(\kappa)$ adds a generic filter for $\mathbb{C}(\kappa);$ where $\mathbb{R}(\kappa)$ and $\mathbb{C}(\kappa)$ are the forcing notions for adding…

In the first part of this paper, we explore the possibility for a very large cardinal $\kappa$ to carry a $\kappa$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model…

We prove the consistency of irr(prod limits_{i<kappa}B_i/D)< prod limits_{i<kappa}irr(B_i)/D, where D is an ultrafilter on kappa and each B_i is a Boolean Algebra. This solves the last problem of this form from the Monk's list of problems,…