## Bulletin of Symbolic Logic

### Proper forcing, cardinal arithmetic, and uncountable linear orders

Justin Tatch Moore

#### 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).

#### Article information

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

Dates
First available in Project Euclid: 9 February 2005

https://projecteuclid.org/euclid.bsl/1107959499

Digital Object Identifier
doi:10.2178/bsl/1107959499

Mathematical Reviews number (MathSciNet)
MR2125149

Zentralblatt MATH identifier
1095.03054

#### Citation

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