Volume 1006 of the series Lecture Notes in Mathematics pp 285303
Proper Forcing and Abelian Groups
 Alan H. MeklerAffiliated withResearch supported by Natural Science, Engineering Council of Canada
Abstract
It is well known that assuming Martin’s axiom and the negation of the continuum hypothesis (MA+−1CH) has a dramatic effect on Abelian group theory. One only has to think of Shelah’s resolution of the Whitehead problem (cf[S1] or [E1]). MA+−NCH has certain drawbacks. It is not strong enough to resolve questions about the structure of ω_{1}separable groups which can be resolved by an extension and it is not consistent with the continuum hypothesis. In this paper I shall talk about generalizations due to Shelah of MA+−ICH, the proper forcing axioms, and their relation to Abelian group theory. (From hereon “group” will mean “Abelian group”.) Section I is devoted to some preliminaries and the statement of the proper forcing axioms. In section II a useful result about choosing a cub almost disjoint from a ladder system assuming PFA(ω_{1}) is proved in order to illustrate how the axiom works. Proper posets are redefined in section III using elementary submodels. This section simplifies the task of writing the remainder of the paper. But it is explained for the reader who is unversed in logic how to ignore this section.
 Title
 Proper Forcing and Abelian Groups
 Book Title
 Abelian Group Theory
 Book Subtitle
 Proceedings of the Conference held at the University of Hawaii, Honolulu, USA, December 28, 1982 – January 4, 1983
 Pages
 pp 285303
 Copyright
 1983
 DOI
 10.1007/9783662215609_14
 Print ISBN
 9783540123354
 Online ISBN
 9783662215609
 Series Title
 Lecture Notes in Mathematics
 Series Volume
 1006
 Series ISSN
 00758434
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 Additional Links
 Topics
 eBook Packages
 Editors

 Rüdiger Göbel ^{(1)}
 Lee Lady ^{(2)}
 Adolf Mader ^{(2)}
 Editor Affiliations

 1. FB 6 — Mathematik, Universität Essen, Gesamthochschule
 2. Department of Mathematics, University of Hawaii
 Authors

 Alan H. Mekler ^{(3)}
 Author Affiliations

 3. Research supported by Natural Science, Engineering Council of Canada, U0075, Canada
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