Bulletin of Symbolic Logic

Proper forcing, cardinal arithmetic, and uncountable linear orders

Justin Tatch Moore

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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).

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Bull. Symbolic Logic, Volume 11, Issue 1 (2005), 51-60.

First available in Project Euclid: 9 February 2005

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Moore, Justin Tatch. Proper forcing, cardinal arithmetic, and uncountable linear orders. Bull. Symbolic Logic 11 (2005), no. 1, 51--60. doi:10.2178/bsl/1107959499. https://projecteuclid.org/euclid.bsl/1107959499

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