Topics in set theory. Lebesgue measurability, large cardinals, forcing axioms, rho-functions
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.
measurable sets of reals in L[\({\mathbb{R}}]\)
scientific article; zbMATH DE number 50322