On the Weak Reflection Principle
On the Weak Reflection Principle
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论弱反射原理
DOI：
10.1090/s0002-9947-2011-05310-9
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发表时间：
2011
影响因子：
1.3
通讯作者：
J. Krueger
中科院分区：
文献类型：
--
作者：
J. Krueger
ω2的弱反射原理，或WRP（ω2），是指Pω1（ω2）的每个静止子集都反射到ω2中的不可数序数。ω2的反射原理，或RP（ω2），是这样的陈述：Pω1（ω2）的每个静止子集反射到ω2中具有共尾性ω1的序数。设κ是一个κ+-超紧基数，且2κ = κ+.然后存在一个强迫偏序集P，它使κ坍缩为ω2，并且P WRP（ω2）<$<$RP（ω2）。本文研究Pω1（ω2）的平稳子集的反射。回想一下，对于不可数序数α，集合S <$Pω1（α）= {x <$α：|X| < ω1}是平稳的，如果对任意函数F：[α] → α，在S中有一个集合B在F下闭。设S ∈ Pω1（ω2）是平稳集.我们说S反射到α，其中α是ω2中的不可数序数，如果S <$Pω1（α）在Pω1（α）中是平稳的。ω2的弱反射原理，或WRP（ω2），是这样的陈述：对于每一个静止集合S <$Pω1（ω2），在ω2中存在一个不可数序数α，使得S反射到α。这个原理是文[1]中介绍的较强弱反射原理的一个特例。陈述WRP（ω2）具有许多有趣的组合后果，包括2 ≤ ω2（[4]，[6]），€（ω2）（[7]），并且ω2 ∩ cof（ω）的每个静态子集反映为ω2中具有共尾性ω1的序数。一个相关的原理是ω2的反射原理，或RP（ω2），它断言Pω1（ω2）的每个静止子集都反射到ω2中具有共尾性ω1的序数。获得WRP（ω2）的标准模型（例如，通过Levy将一个大基数收缩为ω2）都满足RP（ω2）。此外，使用RP（ω2）比使用WRP（ω2）更容易。因此，WRP（ω2）是否蕴涵RP（ω 2）是一个自然的问题，并且这个问题已经公开了一段时间。一个标准的论证表明，如果对于每个平稳集A <$ω2 <$cof（ω），T <$A成立，则对于任何平稳集S <$Pω1（ω2），存在一个平稳集T <$S，它不反映ω2中任何具有共尾性ω的不可数序数。因此，假设存在这样的钻石，WRP（ω2）意味着RP（ω2）。传统上，这种钻石的存在是从GCH中得知的;最近，Shelah [5]证明了它们是21 = ω2的结果。因此，假设21 = ω2，WRP（ω2）意味着RP（ω2）（最初，这个结果在[2]中通过不同的论证得到了证明）。ω2的弱反射原理与弱紧基数等相容。然而，Sakai [3]已经证明了一种局部反射与ZFC是一致的。具体地说，假设GCH和ω1，存在一个一般扩张，其中存在一个平稳集S <$Pω1（ω2），使得S的每个平稳子集反映到ω2中的一个具有共尾性ω的不可数序数。日期：2010年2月。2000年数学学科分类。小学03 E35;中学03 E05。
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 Levy 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.