Abstract. We study combinatorial properties of the partial order (Dense(Q), ⊆).To do that we introduce cardinal invariants p Q , t Q , h Q , s Q , r Q , i Q describing properties of Dense(Q). These invariants satisfyWe compare them with their analogues in the well studied Boolean algebra P(ω)/fin. We show that p Q = p, t Q = t and i Q = i, whereas h Q > h and r Q > r are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of Q. In particular, we show that non(M) = min{|D| :We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah. 0. Introduction. The aim of this paper is to point out the similarities and differences between the structure of P(ω)/fin and the structure of the collection of dense subsets of the rationals. Such research was suggested by A. Blass in [Bl] and initiated by J. Cichoń in [Ci]. The basic object studied here is the set Dense(Q) = {D ⊆ Q : D is dense} ordered by inclusion, in comparison with the structure ([ω] ω , ⊆). Neither one of them is a separative partial order. The separative quotient of ([ω] ω , ⊆) augmented with the least element 0 is the well known Boolean algebra P(ω)/fin. The separative quotient of (Dense(Q), ⊆) with added least element is not a Boolean algebra, but just a lattice, with two dense sets being in the same equivalence class if and only if their symmetric difference is a 2000 Mathematics Subject Classification: 03E17, 03E35, 06E15. Key words and phrases: rational numbers, nowhere dense ideal, distributivity of Boolean algebras, cardinal invariants of the continuum.

Abstract. We study the cardinal invariants of analytic P-ideals, concentrating on the ideal Z of asymptotic density zero. Among other results we prove min{b, cov (N)} ≤ cov * (Z) ≤ max{b, non(N)}.

ResumenEse artículo comienza con una panorámica sobre la proliferación de información propiciada por las tecnologías digitales y sus posibles consecuencias y efectos para la generación de conocimiento. Además de para la comprensión de la complejidad de una realidad que se nos ofrece filtrada y ruidosa en ese alud de información. Sigue con el análisis de cómo esa realidad repercute en la educación formal. Aquí se exploran algunos de los desafíos que han sido formulados tanto para replantear las finalidades de los sistemas educativos, como los modos de aprender y enseñar. A continuación, se presentan diferentes investigaciones realizadas por el grupo de investigación Esbrina. A partir de ellas se da cuenta de cómo estudiantes y docentes están experimentando esos cambios en sus vidas y cómo afectan las relaciones pedagógicas en las que participan. Finalmente, se pone en cuestión algunos aspectos del propio relato. Especialmente el determinismo adaptativo y acrítico que se genera desde los organismos e instituciones, que asumen lo nuevo sin analizar sus consecuencias en los modos de conocer, las relaciones sociales, los enfoques del trabajo y las desigualdades sociales. Lo que nos lleva a concluir urgiendo a la necesidad de debatir y plantear un proyecto de vida en común. Palabras clave: Sociedad de la información, efectos colaterales, vida en común, retos de la educación, restructuración, neoliberalismo Abstract This article begins with an overview of the proliferation of information provided by digital technologies and their potential implications and effects on knowledge generation. In addition to understanding the complexity of a reality that we are offered filtered and noisy in this avalanche of information. It continues with the analysis of how this reality affects formal education. Here we explore some of the challenges that have been formulated both to rethink the aims of educational systems and the ways of learning and teaching. The following is a presentation of different research projects carried out by the Esbrina research group. From them, we see how students and teachers are experiencing these changes in their lives and how they affect the pedagogical relationships in which they participate. Finally, some aspects of the narrative itself are questioned. Especially the adaptive and uncritical determinism that is generated from the organisms and institutions, which assume the new without analysing its consequences in the ways of knowing, social relations, work approaches and social inequalities. This leads us to conclude by urging us to debate and propose a project of common life.

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