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),∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω1, 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 κ unless stationary subsets of Sκ+ω reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω2).
Citation
Justin Tatch Moore. "Proper forcing, cardinal arithmetic, and uncountable linear orders." Bull. Symbolic Logic 11 (1) 51 - 60, March 2005. https://doi.org/10.2178/bsl/1107959499
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