Related papers: Covering of the null ideal may have countable cofi…

We show that the class of completely m-full ideals coincides with the class of componentwise linear ideals in a polynomial ring over an infinite field.

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…

We consider smooth completion of algebraic manifolds. Having some information about its singular completions or about completions of its images we prove purity of cohohomology of the set at infinity. We deduce also some topological…

We prove the existence of infinite dense free sets (in the usual topology) for set mappings on the reals, under reasonable assumptions.

Let $S$ and $\mathcal{C}$ be affine semigroups in $\mathbb{N}^d$ such that $S\subseteq \mathcal{C}$. We provide a characterization for the set $\mathcal{C}\setminus S$ to be finite, together with a procedure and computational tools to check…

We present necessary and sufficient conditions for the existence of a countably additive measure on a complete Boolean algebra.

We show that all integrally closed ideals on log terminal surfaces are multiplier ideals by extending an existing proof for smooth surfaces.

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…

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(meagre_lambda), i.e. the minimal number of nowhere dense subsets of…

We prove that in some cases definable chains of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic chains, ROD chains in the Solovay model, and $\Sigma^1_2$ chains in the assumption that…

We characterize which complete local (Noetherian) rings T containing the rationals are the completion of a countable excellent local ring S. We also discuss the possibilities for the map from the minimal prime ideals of T to the minimal…

We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support…

We prove that the set of proper ideals of a monoid endowed with coarse lower topology is a spectral space.

Cantor's famous proof of the non-denumerability of real numbers does apply to any infinite set. The set of exclusively all natural numbers does not exist. This shows that the concept of countability is not well defined. There remains no…

We give simple necessary and sufficient conditions for the inclusion-exclusion identity to hold for an infinite countable number of sets. In terms of a random variable, whose range are nonnegative integers, this condition is equivalent to…

Let $\mathcal{E}$ denote the $\sigma$-ideal generated by closed null sets on the reals. We show that the uniformity and the covering of $\mathcal{E}$ can be added to Cicho\'n's maximum with distinct values. More specifically, it is…

We investigate the connection between measure and capacity for the space of nonempty closed subsets of {0,1}*. For any computable measure, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets…

Effective versions of strong measure zero sets are developed for various levels of complexity and computability. It is shown that the sets can be equivalently defined using a generalization of supermartingales called odds supermartingales,…

In standard measure theory the measure on the base set Omega is normalised to one, which encodes the statement that "Omega happens". Moreover, the rules imply that the measure of any subset A of Omega is strictly positive if and only if A…

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we…