Related papers: Supersaturated ideals

If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms introduced by Hamkins.

We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{\omega_1}+$all sets of reals are $I_{\mathbb Q,\aleph_0}$-measurable'' implies the existence of an inner model with a measurable…

Let $\mathscr{F}=(F_n)$ be a sequence of nonempty finite subsets of $\omega$ such that $\lim_n |F_n|=\infty$ and define the ideal $$\mathcal{I}(\mathscr{F}):=\left\{A\subseteq \omega: |A\cap F_n|/|F_n|\to 0~\mbox{as}~n\to \infty \right\}.$$…

I introduce a new family of axioms extending ZFC set theory, the $\Sigma_n$-correct forcing axioms. These assert roughly that whenever a forcing name $\dot{a}$ can be forced by a poset in some forcing class $\Gamma$ to have some $\Sigma_n$…

In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a…

Given a compact Polish space $E$ and the hyperspace of its compact subsets $\mathcal{K}(E)$, we consider the class of $G_{\delta}$ $\sigma$-ideals of compact subsets of $E$ that can be represented via a compact subset of $\mathcal{K}(E)$.…

We study $\Sigma_1(\omega_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $\Sigma_1$-formula with parameter $\omega_1$) in the presence of large cardinals. Our results show that the existence…

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the…

We introduce a unified framework for studying persistence phenomena in commutative algebra via filtrations of ideals. For a filtration $\mathcal{F} = \{I_i\}_{i \in \mathbb{N}}$, we define $\mathcal{F}$-persistence and $\mathcal{F}$-strong…

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the…

Let $G$ consist of all functions $g \colon \omega \to [0,\infty)$ with $g(n) \to \infty$ and $\frac{n}{g(n)} \nrightarrow 0$. Then for each $g\in G$ the family $\mathcal{Z}_g=\{A\subseteq\omega:\ \lim_{n\to\infty}\frac{\text{card}(A\cap…

Let $X$ be a first countable space which admits a non-trivial convergent sequence and let $\mathcal{I}$ be an analytic P-ideal. First, it is shown that the sets of $\mathcal{I}$-limit points of all sequences in $X$ are closed if and only if…

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every…

We ask whether $\mathbf{\Delta^1_2}$ or $\mathbf{\Sigma^1_2}$ equivalence relations with $I$-small classes for $I$ a $\sigma$-ideal must have perfectly many classes. We show that for a wide class of ccc $\sigma$-ideals, a positive answer…

Let $\Sigma (X,\mathbb{C})$ denote the collection of all the rings between $C^*(X,\mathbb{C})$ and $C(X,\mathbb{C})$. We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal…

A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…

We study models M of set theory that are "condensable", in the sense that there is an "ordinal" v of M such that the rank initial segment of M determined by v is both isomorphic to M, and also an elementary submodel of M for infinitary…

In this article, we continue the studying of $\mathcal{H}_Y$-ideals. We introducing two notions fixed and free $\mathcal{H}_Y$-ideals as an extension of fixed and free z-ideals in C(X) and relative $\mathcal{H}_Y$-ideals as an extension of…

If $S$ is a discrete semigroup, then $\beta S$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $\beta S$, and…