On the Weak Reflection Principle

The Weak Reflection Principle for ω2, or WRP(ω2), is the statement that every stationary subset of Pω1 (ω2) reflects to an uncountable ordinal in ω2. The Reflection Principle for ω2, or RP(ω2), is the statement that every stationary subset of Pω1 (ω2) reflects to an ordinal in ω2 with cofinality ω1. Let κ be a κ+-supercompact cardinal and assume 2κ = κ+. Then there exists a forcing poset P which collapses κ to become ω2, and P WRP(ω2)∧¬RP(ω2). In this paper we will be concerned with reflection of stationary subsets of Pω1(ω2). Recall that for an uncountable ordinal α, a set S ⊆ Pω1(α) = {x ⊆ α : |x| < ω1} is stationary if for any function F : [α] → α, there is a set b in S which is closed under F . Let S ⊆ Pω1(ω2) be a stationary set. We say that S reflects to α, where α is an uncountable ordinal in ω2, if S ∩ Pω1(α) is stationary in Pω1(α). The Weak Reflection Principle for ω2, or WRP(ω2), is the statement that for every stationary set S ⊆ Pω1(ω2), there is an uncountable ordinal α in ω2 such that S reflects to α. This principle is a special case of the stronger Weak Reflection Principle introduced in [1]. The statement WRP(ω2) has a number of interesting combinatorial consequences, including 2 ≤ ω2 ([4], [6]), ¬ (ω2) ([7]), and every stationary subset of ω2 ∩ cof(ω) reflects to an ordinal in ω2 with cofinality ω1. A related principle is the Reflection Principle for ω2, or RP(ω2), which asserts that every stationary subset of Pω1(ω2) reflects to an ordinal in ω2 with cofinality ω1. The standard models for obtaining WRP(ω2) (for example, by Lévy collapsing a large cardinal to become ω2) all satisfy RP(ω2). Also, it tends to be easier to work with RP(ω2) than with WRP(ω2). Thus it is a natural question, and one which has been open for some time, whether WRP(ω2) implies RP(ω2). A standard argument shows that if ♦A holds for every stationary set A ⊆ ω2 ∩ cof(ω), then for any stationary set S ⊆ Pω1(ω2), there is a stationary set T ⊆ S which does not reflect to any uncountable ordinal in ω2 with cofinality ω. So assuming the existence of such diamonds, WRP(ω2) implies RP(ω2). Classically, the existence of such diamonds were known to follow from GCH; more recently, Shelah [5] has proven they are a consequence of 21 = ω2. Thus assuming 21 = ω2, WRP(ω2) implies RP(ω2) (originally, this result was proven in [2] by a different argument). The Weak Reflection Principle for ω2 is equiconsistent with a weakly compact cardinal. However, Sakai [3] has shown that a kind of local reflection is consistent from ZFC. Specifically, assuming GCH and ω1 , there is a generic extension in which there exists a stationary set S ⊆ Pω1(ω2) such that every stationary subset of S reflects to an uncountable ordinal in ω2 with cofinality ω. Date: February 2010. 2000 Mathematics Subject Classification. Primary 03E35; Secondary 03E05.