Perfect-set Forcing for Uncountable Cardinals
Perfect-set Forcing for Uncountable Cardinals
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不可数基数的完美集强迫
DOI：
10.1016/0003-4843(80)90021-2
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
I. AkihiroKANAMOR
中科院分区：
文献类型：
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作者：
I. AkihiroKANAMOR
完美设定的强迫已经存在很长时间了。萨克斯本人在集合论和递归理论中都大量地利用它得到了重要的极小性结果，他推广的融合思想已经成为强迫的几个概念的组成部分。在Laver[8]发展了实数与可数支集迭代相加的思想之后，Baumgartner和Laver[2]将其应用于完美集强迫的情形，得到了关于环上的Ramsey超滤子和CO2的树性的有趣的一致性结果。此后，Shelah，Baumgartner等人的工作相当系统化了可数支集迭代强迫。作为推广的第一步，我在本文中提出了正则不可数基数K的完美集强制的概念及其K大小支集的迭代。在Sack和Slaman[11]最近的工作中，这种强迫的一个有效版本已经在研究抽象E-递归和E-闭结构的侧向扩张中得到了应用。第一节阐述了强迫及其迭代的概念，并建立了它们的基本性质。特别地，提出并证明了适当的融合引理。第二节主要是一个关键技术定理的长篇证明，它的许多结果之一是~+被迭代强迫保持为基数。在地面模型中使用~序列是这种聚变论点的一个基本特征。与文献[2]中所考虑的TO情形相比，在不可数情形下对强迫机制的控制要少得多，但是<；>；K给了我们关于K的子集的足够的结构信息，以允许更经济的程序工作。事实上，很明显，这篇论文要归功于[2]。新的调制主要来自极限阶段的构造和O~的使用。第三节证明了如果2~=~*，则强迫的~<；K~+迭代仍保持·*‘，但一般情况下，K~迭代加a<；~.序列(实际上是K*-Suslin树)，因此如果地面模型中已满足2~&K+，则K H折叠。在第四节中，提升了[2]中关于Aronszajn树的结果：使用<；~.建立了迭代强迫的闭包性质，这意味着，正如Silver首先在Mitchell模型(见[9])中所表明的那样，如果强迫迭代h次，其中A是一个弱紧基数>；K，则在所得到的扩张中不存在K++-Aronszajn树。
Perfect-set forcing has been around for a long time. Sacks [10] himself had made substantial use of it to get important minimality results both in set theory and in recursion theory, and the fusion idea that he popularized has become an integral part of several notions of forcing. After Laver [8] developed the idea of adding reals iteratively with countable support, Baumgartner and Laver [2] applied it to the case of perfect-set forcing to produce interesting consistency results about Ramsey ultrafilters over to and the tree property for co 2. Since then, work of Shelah, Baumgartner, and others has considerably systematized countable support iterated forcing. As a first step in generalization, I develop in this paper a notion of perfect-set forcing for regular uncountable cardinals K and its iteration with K size supports. An application of an effective version of this forcing has already been made in recent work by Sacks and Slaman [11] in the study of abstract E-recursion and sideways extensions of E-closed structures. in Section I the notion of forcing and its iteration are formulated, and their basic properties established. In particular, the appropriate fusion lemmas are stated and proved. Section 2 is dominated by the long proof of a key technical theorem, one of whose many consequences is that ~ + is preserved as a cardinal by the iterated forcing. The use of a ~ sequence in the ground model is an essential feature of this fusion argument. There is much less control over the forcing machinery in the uncountable case as compared to the to case considered in [2], but <>K gives us just enough structural information about subsets of K to allow more economical procedures to work. In fact, it will be clear that this paper owes an obvious debt to [2]. with the new modulations arising primarily from limit stage constructions and the use of O~. in Section 3 it is shown that if 2 ~ = ~*, then ~<K ~+ iterations of the forcing still preserves •*' , hut that, in general, K ~ iterations adds a <~. sequence (in fact, a K*-Suslin tree) and hence collapses K H if 2~>K + had been satisfied in the ground model. In Section 4, the result on Aronszajn trees in [2] is lifted: Using <~. a closure property for the iterated forcing is established, and this implies, as Silver first showed in Mitchell's model (see [9]), that if the forcing is iterated h times, where A is a weakly compact cardinal >K, then there are no K++-Aronszajn trees in the resulting extension.