Proper Forcing, Cardinal Arithmetic, and Uncountable Linear Orders

Abstract

AbstractIn 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 areX, ω1,,C,C *whereXis any suborder of the reals of size ω1andCis any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis atkunless stationary subsets ofreflect. The techniques are expected to be applicable to other open problems concerning the theory ofH(ω2).