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Proper Forcing, Cardinal Arithmetic, and Uncountable Linear Orders

Published online by Cambridge University Press:  15 January 2014

Justin Tatch Moore
Justin Tatch Moore*
Affiliation:
Department of Mathematics, Boise State University, Boise, Idaho 83725, USA E-mail: justin@math.boisestate.edu
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Abstract

In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of ℝ which is Σ1-definable in (H(ω2), ϵ). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω1, , C, C * where X is any suborder of the reals of size ω1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at k unless stationary subsets of reflect. The techniques are expected to be applicable to other open problems concerning the theory ofH(ω2).

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Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 11 , Issue 1 , March 2005 , pp. 51 - 60
Copyright
Copyright © Association for Symbolic Logic 2005

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