Several types of polarized partition relations are considered. In particular we deal with partitions defined on cartesian products of more than two factors. MSC: 03E05.

Given a topological Ramsey space math formula, we extend the notion of semiselective coideal to sets math formula and study conditions for math formula that will enable us to make the structure math formula a Ramsey space and also study forcing notions related to math formula which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space. This extends results from to the most general context of topological Ramsey spaces. As applications, we (...) prove that for every topological Ramsey space math formula, under suitable large cardinal hypotheses every semiselective ultrafilter math formula is generic over math formula; and that given a semiselective coideal math formula, every definable subset of math formula is math formula-Ramsey. This generalizes the corresponding results for the case when math formula is equal to Ellentuck's space. (shrink)

The Continuum Hypothesis has motivated a considerable part of the development of axiomatic set theory for over a century. We present, in a very schematic way, some of the results that give information related to Cantor’s Continuum Problem.

We prove that the least σ-algebra containing the projective sets and closed under projections is exactly the collection of hyperprojective sets which, with their complements, can be inductively defined with real parameters by an induction of countable length. This provides a construction principle for this natural class of hyperprojective sets.

We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition (...) of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. (shrink)

We study several perfect set properties of the Baire space which follow from the Ramsey property ω→(ω) ω . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

El objetivo de este trabajo es presentar en un solo cuerpo tres maneras de relativizar (o generalizar) el concepto de conjunto constructible de Gödel que no suelen aparecer juntas en la literatura especializada y que son importantes en la Teoría de Conjuntos, por ejemplo para resolver problemas de consistencia o independencia. Presentamos algunos modelos resultantes de las diferentes formas de relativizar el concepto de constructibilidad, sus propiedades básicas y algunas formas débiles del Axioma de Elección válidas o no válidas en (...) ellas. (shrink)

The Principle of Ariadne, formulated in 1988 ago by Walter Carnielli and Carlos Di Prisco and later published in 1993, is an infinitary principle that is independent of the Axiom of Choice in ZF, although it can be consistently added to the remaining ZF axioms. The present paper surveys, and motivates, the foundational importance of the Principle of Ariadne and proposes the Ariadne Game, showing that the Principle of Ariadne, corresponds precisely to a winning strategy for the Ariadne (...) Game. Some relations to other alternative. set-theoretical principles are also briefly discussed. (shrink)

Este artigo tem como objetivo abordar o problema das relações objetais primárias recorrendo a alguns conceitos formulados por Michael Balint. Para isto, parte-se da discussão sobre duas posições ou modos de investimento precoces, para em seguida situá-los no contexto da falha básica, compreendida como um mecanismo primitivo de constituição da subjetividade e uma das principais áreas do aparato psíquico. Com isso pretende-se oferecer um esboço da teoria balintiana sobre as origens da subjetivação, articulando-a com a chamada clínica de pacientes difíceis.This (...) article aims to investigate the problem of primary object relations appealing to some of the concepts formulated by Michael Balint. In this sense, it starts with the discussion of two positions or modes of precocious investment, which are then situated in the context of the basic fault, understood as a primitive mechanism in subjectivity's constitution and as one of the main areas of the psychic apparatus. In this way, it intends to offer a sketch of Balint's theory about the origins of subjectivity, articulated with the so-called clinic of difficult patients. (shrink)

Seekers know that they will search all their lives not only for what they have lost, but above all for what they do not yet know and that they are not even sure exists. Augusto Cavadi, to whom this book is dedicated on his 70th birthday, is an indomitable seeker. For this reason a group of friends and colleagues wrote these collected testimonies, reflections, ideas and suggestions inspired by his writings and his life as an anti-academic philosopher without a (...) tie--Translated from back cover. (shrink)