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Forcings constructed along morasses

Published online by Cambridge University Press:  12 March 2014

Bernhard Irrgang
Bernhard Irrgang*
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany, E-mail: bernhard.irrgang@googlemail.com
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Abstract

We further develop a previously introduced method of constructing forcing notions with the help of morasses. There are two new results: (1) If there is a simplified (ω1, 1)-morass, then there exists a ccc forcing of size ω1 that adds an ω2-Suslin tree. (2) If there is a simplified (ω1, 2)-morass, then there exists a ccc forcing of size ω1 that adds a 0-dimensional Hausdorff topology τ on ω3 which has spread s(τ) = ω1. While (2) is the main result of the paper, (1) is only an improvement of a previous result, which is based on a simple observation. Both forcings preserve GCH. To show that the method can be changed to produce models where CH fails, we give an alternative construction of Koszmider's model in which there is a chain 〈Xα ∣ α < ω2〉 such that Xα ⊆ ω1. Xβ – Xα is finite and Xα – Xβ has size ω1 for all β < α < ω2.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1097 - 1125
Copyright
Copyright © Association for Symbolic Logic 2011

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