Topics in Set Theory Lebesgue Measurability, Large Cardinals, Forcin
During the Fall Semester of 1987, Stevo Todorcevic gave a series of lectures at the University of Colorado. These notes of the course, taken by the author, give a novel and fast exposition of four chapters of Set Theory. The first two chapters are about t
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
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M. Bekkali
Topics in Set Theory Lebesgue Measurability, Large Cardinals, Forcing Axioms, Rho-functions
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Author Mohamed Bekkali Department of Mathematics University of Colorado P. O. Box 426 Boulder, CO 80309-0426, USA
Mathematics Subject Classification (1980): 04-02, 03E05, 03E55
ISBN 3-540-54121-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54121-7 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
During the Fall 1987 Professor Stevo Todorcevic gave a series of lectures for the Ulam Seminar at the University of Colorado at Boulder. This is a reworked version of my handwritten notes taken during the Seminar. I have added a few supplementary results found in the well-known texts listed in the references and added a few details in some of the proofs, but the basic sequence of results remains unchanged. Since the set of notes lacked any historical discussion, only a partial list of references has been supplied at the end of each chapter. I would like to thank Professor Stevo Todorcevic for a large amount of time spent after the in explaining the proofs to me and for the permission of including here a number of his unpublished notes. This work also greatly depended on Liz Stimmel who, with utmost patience and care, typed the manuscript.
M. Bekkali Boulder; January 1991
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Contents Chapter 1: Nonmeasurable Sets of Reals
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This chapter deals with various examples of nonmeasurable sets of reels. Also a definable (in fact nonmeasurable set of reels is given assuming that WI is not inaccessible in L. In other words if all projective sets of reals are measurable, then WI is an inaccessible cardinal in L. Chapter 2: Measurability in L[m]
15-22
The assumption WI is not inaccessible in L, in the previous chapter, is a too strong restriction on the universe of sets. So, it cannot be considered as an answer to the old problem: Does there exist a definable nonmeasurable set of reels? By contrast with measurable cardinals, we'll see that supercompact cardinals settle this problem completely. The main ingredient in proving this result is the notion of a saturated ideal over WI. 23-60
Chapter 3: Fo
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