Topics in set theory. Lebesgue measurability, large cardinals, forcing axioms, rho-functions
type: Item
instance of: scholarly article
title: Topics in set theory. Lebesgue measurability, large cardinals, forcing axioms, rho-functions
zbMATH Open document ID: 0729.03022
author: Mohamed Bekkali
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: This book consists of a reworked version of notes taken by the author from a series of lectures given by S. Todorcevic in Fall 1987 for the Ulam Seminar at the University of Colorado at Boulder. There are four chapters (and an appendix). The first chapter deals with various examples of non-Lebesgue measurable sets of reals, including the result that if all \(\Sigma^ 1_ 3\) sets of reals are Lebesgue measurable, then \(\omega_ 1\) is an inaccessible cardinal in L. Chapter two considers measurable sets of reals in L[\({\mathbb{R}}]\). The main result of the chapter is that if there is a supercompact cardinal, then every set of reals in L[\({\mathbb{R}}]\) is Lebesgue measurable. The proof is based on saturated ideals over \(\omega_ 1.\) Chapter three is devoted to forcing axioms. The first part is spent on Martin's axiom, characterizing Martin's axiom in terms of partitions. In particular, it is shown that \(MA_{\aleph_ 1}\) is equivalent to the statement that every ccc-partition \([\omega_ 1]^{<\omega}=K_ 0\cup K_ 1\) has an uncountable homogeneous set. Chapter four contains a detailed exposition of Todorcevic's method of minimal walks. [See \textit{S. Todorcevic}, Acta Math. 159, 261-294 (1987; Zbl 0658.03028).] Applications are given to square sequences, trees and partition relations.
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E65
zbMATH DE Number: 50322
zbMATH Keywords: non-Lebesgue measurable sets of reals
zbMATH Keywords: inaccessible cardinal
zbMATH Keywords: measurable sets of reals in L[\({\mathbb{R}}]\)
zbMATH Keywords: supercompact cardinal
zbMATH Keywords: saturated ideals
zbMATH Keywords: forcing axioms
zbMATH Keywords: Martin's axiom
zbMATH Keywords: partitions
zbMATH Keywords: uncountable homogeneous set
zbMATH Keywords: minimal walks
zbMATH Keywords: square sequences
zbMATH Keywords: trees
zbMATH Keywords: partition relations
reviewed by: Neil H. Williams
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 50322