Internally club and approachable for larger structures
Internally club and approachable for larger structures
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内部俱乐部，适合大型结构
DOI：
10.4064/fm201-2-2
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
J. Krueger
中科院分区：
文献类型：
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作者：
J. Krueger
我们将正则基数κ的脂肪子集的概念推广到Pκ(X)的脂肪子集，其中κ≥X，设μ< κ, μ<μ = μ，且κ是超紧的。然后存在一个κ = μ++的一般扩展，并且对于所有正则λ≥μ++，在[H(λ)]μ中存在平稳的多个内部俱乐部但内部不可接近的N。假设μ是一个无限基数。如果N是由大小为μ的集合组成的递增连续序列< Ni: i < μ >的并集，使得对于所有α < μ, < Ni: i < α >在N内，则集合N是长度为μ的内部可接近的。一个相关的想法是内部俱乐部设置。如果N∩[N]包含[N]的俱乐部子集，则大小为μ的集合N是内部俱乐部。换句话说，N是一个递增的连续序列< Ni: i < μ >的并集，其大小为μ，使得每个Ni都在N中。福尔曼和托多切维奇的问题是，内部平易近人和内部俱乐部的属性是否相等。在[5]中，我们证明了在PFA条件下，对于所有正则λ≥ω2，存在许多大小为φ 1的结构N <s:1> H(λ)，使得N是内部俱乐部的，但不是内部可接近的。本文将这一结果推广到更大的结构中。定理1。假设μ < κ, μ = μ，且κ是超紧的。然后存在一个使κ坍缩为μ的μ闭、μ固有强迫偏序集，并强迫对于所有正则λ≥μ，在[H(λ)]μ+中存在平稳的多个内部俱乐部但内部不可接近的N。在我们用来证明定理1的模型中，我们有2 = μ。事实上，如果2 = μ，则任何大小为μ且包含μ的初等子结构N * H（λ）如果是内部可接近的，则它是内部棍棒的；这在论文的最后显示。在第1节中，我们将回顾符号和一些背景材料。第2节将正则基数κ的胖子集的思想推广到Pκ(X)的胖子集，其中κ任X。第3节给出了我们在一致性结果中使用的基本强迫偏序集，并在第4节中描述了如何使用混合支持强迫迭代迭代该偏序集。在第5节中，我们证明定理1。
We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Suppose μ < κ, μ<μ = μ, and κ is supercompact. Then there is a generic extension in which κ = μ++, and for all regular λ ≥ μ++, there are stationarily many N in [H(λ)]μ which are internally club but not internally approachable. Suppose μ is an infinite cardinal. A set N is internally approachable with length μ if N is the union of an increasing and continuous sequence 〈Ni : i < μ〉 of sets with size μ such that for all α < μ, 〈Ni : i < α〉 is in N . A related idea is that of an internally club set. A set N with size μ is internally club if N ∩ [N ] contains a club subset of [N ]. In other words, N is the union of an increasing and continuous sequence 〈Ni : i < μ〉 of sets with size μ such that each Ni is in N . Foreman and Todorcevic [3] asked whether the properties of being internally approachable and internally club are equivalent. In [5] we proved that under PFA, for all regular λ ≥ ω2 there are stationarily many structures N ≺ H(λ) with size א1 such that N is internally club but not internally approachable. In this paper we generalize this result to larger structures. Theorem 1. Suppose μ < κ, μ = μ, and κ is supercompact. Then there is a μ-closed, μ-proper forcing poset which collapses κ to become μ, and forces that for all regular λ ≥ μ, there are stationarily many N in [H(λ)]μ+ which are internally club but not internally approachable. In the model we construct to prove Theorem 1, we have that 2 = μ. In fact, if 2 = μ, then any elementary substructure N ≺ H(λ) with size μ and which contains μ is internally club iff it is internally approachable; this is shown at the end of the paper. In Section 1 we review notation and some background material. Section 2 generalizes the idea of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Section 3 presents the basic forcing poset we use in our consistency result, and in Section 4 we describe how to iterate this poset with a mixed support forcing iteration. In Section 5 we prove Theorem 1.