Basis problems in combinatorial set theory
type: Item
instance of: scholarly article
title: Basis problems in combinatorial set theory
zbMATH Open document ID: 0918.03025
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/228251
full work available at URL: Todorcevic.MAN.html
review text: The author discusses results and conjectures concerning some basis problems in set theory. A basis for a class \(S\) of structures is a collection \(S_0\) of critical members of \(S\) such that any given structure from \(S\) can be related via some kind of connecting map to one member from \(S_0\). For instance, the author showed that under PFA, the five sets 1, \(\omega\), \(\omega_1\), \(\omega\times \omega_1\) and \(\text{Fin}_{\omega_1}\) (the collection of all finite subsets of \(\omega_1\) ordered by inclusion) form a basis for the class of all directed sets on \(\omega_1\) in the sense that any such set is Tukey equivalent to one of the five. The author inspects basis problems in a variety of areas, considering successively binary relations on \(\omega_1\), transitive relations on \(\omega_1\), uncountable linear orderings and uncountable regular spaces. He also discusses related Ramsey-theoretic principles such as the Open Coloring Axiom or special Aronszajn trees.
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03E65
Mathematics Subject Classification ID: 54E45
zbMATH DE Number: 1184262
zbMATH Keywords: combinatorial set theory
zbMATH Keywords: Proper Forcing Axiom
zbMATH Keywords: Ramsey theory
zbMATH Keywords: research survey
zbMATH Keywords: basis
zbMATH Keywords: directed sets
zbMATH Keywords: successively binary relations
zbMATH Keywords: transitive relations
zbMATH Keywords: uncountable linear orderings
zbMATH Keywords: uncountable regular spaces
zbMATH Keywords: Open Coloring Axiom
zbMATH Keywords: Aronszajn trees
author: Stevo Todorčević
reviewed by: Pierre Matet
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184262