Nous présentons dans ce papier une définition purement syntaxique des types entrées et des types sorties du système $ℱ$. Nous définissons les types de données syntaxiques comme étant des types entrées et sorties. Nous démontrons que les types à quantificateurs positifs sont des types de données syntaxiques et qu’un type entrée est un type sortie. Nous imposons des restrictions sur la règle d’élimination des quantificateurs pour démontrer qu’un type sortie est un type entrée.

We give in this paper a purely syntactical definition of input and output types of system $ℱ$. We define the syntactical data types as input and output types. We show that any type with positive quantifiers is a syntactical data type and that an input type is an output type. We give some restrictions on the ∀-elimination rule in order to prove that an output type is an input type.

We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from $cof\left(\right)<2^{\omega }$) that less than $2^{\omega }$ many translates of a compact set of measure zero can cover ℝ.

In his paper [Kybernetika 31, No. 1, 99–106 (1995; Zbl 0857.03042)], E. Turunen says in the corollary on p. 106: “Notice that the third last line on page 195 in [J. K. Mattila, “Modifier logic”, in: J. Kacprzyk (ed.) et al., Fuzzy logic for the management of uncertainty. New York: Wiley. 191–209 (1992)] stating that LPC+Ch calculus is consistent is not correct.” The system LPC+Ch is consistent, which can be seen quite trivially.

We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures $P_{\kappa }\left(\lambda \right)$ carries. In the first of these models, $P_{\kappa }\left(\lambda \right)$ carries $2^{2^{{\left[\lambda \right]}^{<\kappa }}}$ many normal measures, the maximal number. In the second of these models, $P_{\kappa }\left(\lambda \right)$ carries $2^{2^{{\left[\lambda \right]}^{<\kappa }}}$ many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and hence also $P_{\kappa }\left(\kappa \right)$)...

We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.

We introduce the notion of leveled structure and show that every structure elementarily equivalent to the real expo field expanded by all restricted analytic functions is leveled.

Let ${𝒯}_n$ denote the set of log canonical thresholds of pairs $(X,Y)$, with $X$ a nonsingular variety of dimension $n$, and $Y$ a nonempty closed subscheme of $X$. Using non-standard methods, we show that every limit of a decreasing sequence in ${𝒯}_n$ lies in ${𝒯}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that ${𝒯}_n$ is closed in $𝐑$; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check...

We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{\mathbb{Z}\left[t\right]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{\mathbb{Z}\left[t\right]}$ containing $G$ is universally equivalent to $G$. Since finitely generated...

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{\aleph ₀}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are $G_{\delta }$ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property,...