In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that [Formula: see text] satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □ fails for all regular κ > ω1.

The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at κ then every stationary subset of reflects. It will also be demonstrated (...) that MRP always fails in a generic extension by Prikry forcing. (shrink)

In this paper we will analyze Baumgartner’s problem asking whether it is consistent that \ and every pair of \-dense subsets of \ are isomorphic as linear orders. The main result is the isolation of a combinatorial principle \\) which is immune to c.c.c. forcing and which in the presence of \ implies that two \-dense sets of reals can be forced to be isomorphic via a c.c.c. poset. Also, it will be shown that it is relatively consistent with ZFC (...) that there exists an \ dense suborder X of \ which cannot be embedded into \ in any outer model with the same \. (shrink)

We prove that there is a -ideal of compact sets which is strictly above in the Tukey order. Here is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličković asked in [Alain Louveau, Boban Veličković, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 171–195].

The purpose of this note is to demonstrate that a weak form of club guessing on ω1 implies the existence of an Aronszajn line with no Countryman suborders. An immediate consequence is that the existence of a five element basis for the uncountable linear orders does not follow from the forcing axiom for ω-proper forcings.

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 Σ 1 -definable in (H(ω 2 ), ∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω 1 , ω 1 * , C, C * where X is any suborder of the reals of size (...) ω 1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at κ unless stationary subsets of S κ + ω reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω 2 ). (shrink)

The purpose of this article is to prove the relative consistency of certain statements about open colorings with 2ℵ0 < 2ℵ1. In particular both OCA and the statement that every 1–1 function of size ℵ1 is σ-monotonic are consistent with 2ℵ0 < 2ℵ1. As a corollary we have that 2ℵ0 < 2ℵ1 does not admit a ℙ max variation.