We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.

We formulate a Covering Property Axiom $CP{A}_{cube}$, which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨fₙ\in {ℝ}^{ℝ}{⟩}_{n<\omega }$ of Borel functions there are sequences:...

For any three noncollinear points c₀,c₁,c₂ ∈ ℝ², there are sprays S₀,S₁,S₂ centered at c₀,c₁,c₂ that cover ℝ². This improves the result of de la Vega in which c₀,c₁,c₂ were required to be the vertices of an equilateral triangle.

We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

We develop the theory of topological Hurewicz test pairs: a concept which allows us to distinguish the classes of the Borel hierarchy by Baire category in a suitable topology. As an application we show that for every ${\Pi }_{\xi }^0$ and not ${\Sigma }_{\xi }^0$ subset $P$ of a Polish space $X$ there is a $\sigma$-ideal $ℐ\subseteq 2^X$ such that $P\notin ℐ$ but for every ${\Sigma }_{\xi }^0$ set $B\subseteq P$ there is a ${\Pi }_{\xi }^0$ set $B^{\text{'}}\subseteq P$ satisfying $B\subseteq B^{\text{'}}\in ℐ$. We also discuss several other results and problems related to ideal generation and Hurewicz test pairs.

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ${}^{\omega }\omega$ of irrationals, or certain of its subspaces. In particular, given $f\in {}^{\omega }(\omega \setminus \left\{0\right\})$, we consider compact sets of the form ${\prod }_{i\in \omega }{B}_i$, where $|B_i|=f\left(i\right)$ for all, or for infinitely many, $i$. We also consider “$n$-splitting” compact sets, i.e., compact sets $K$ such that for any $f\in K$ and $i\in \omega$, $\left|\right\{g\left(i\right):g\in K,g\upharpoonright i=f\upharpoonright i\left\}\right|=n$.

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.

Si espongono alcuni risultati, provati dall’Autore negli articoli citati nella bibliografia, a proposito della complessità del teorema d’interpolazione di Craig: con ciò si intende la relazione tra la lunghezza (cioè il numero di simboli) della formula $\chi$ e la lunghezza di $\phi$ e $\psi$, ove $\phi \to \psi$ è un’implicazione valida, e $\chi$ è un interpolante, come esibito dal teorema di interpolazione stesso. Si intende altresì sottolineare la rilevanza dello studio della complessità dell’interpolazione per far luce su alcuni...

In [6] da Costa has introduced a new hierarchy $N_i$, $1\le i\le w$ of logics that are both paraconsistent and paracomplete. Such logics are now known as non-alethic logics. In this article we present an algebraic version of the logics $N_i$ and study some of their properties.

The resemblance relation is used to reflect some real life situations for which a fuzzy equivalence is not suitable. We study the properties of cuts for such relations. In the case of a resemblance on a real line $E$ we show that it determines a special family of crisp functions closely connected to its cut relations. Conversely, we present conditions which should be satisfied by a collection of real functions in $E$ in order that this collection determines a resemblance relation.

We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among finite graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counterpart of this fact for all bipartite graphs in the class of all graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the...