RUI: Outer Models and Forcing

RUI：外部模型和强迫

• 批准号: 9803643
• 负责人: Maurice Stanley
• 金额: $ 8.57万
• 依托单位: San Jose State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803643&HistoricalAwards=false
• 关键词: RUI; Outer; Models; Forcing

Stanley proposes to continue his work in set theory in two areas, namely, infinitary combinatorics and class forcing. One group of combinatorial problems concerns characterizing which stationary sets can contain closed unbounded subsets in cardinal and GCH preserving outer models. Every stationary subset of aleph-one has this property. Previous work has shown that in general there is no such first-order characterization for subsets of aleph-two. On the other hand, for a restricted class of subsets of the successor of aleph-omega, such a characterization is possible. Some interesting cases remain open. Similar characterization questions can be asked regarding partitions and trees. Another area for continued work is the study of outer models and class forcing. Notwithstanding some technical qualifications, essentially only one method is known for constructing outer models, namely, Cohen's forcing method. Work to this point has shown that this is not an accident: Granted a technical assumption, every outer model is a class forcing extension, in one sense of the notion. An open problem is to characterize outer models that are class forcing extensions in a certain stronger sense. One point of this work is reducing certain set theoretical questions to more concrete combinatorial questions. All mathematical statements can be translated into statements about sets. If a given statement is mathematically provable, then the corresponding statement about sets is derivable from the axioms of set theory. Thus statements whose translations are neither provable nor refutable from the axioms of set theory are themselves neither provable nor refutable mathematically. Such statements are said to be "independent". In some cases, a given mathematical object may have a certain property for provable reasons; in others, for independent reasons. One subject of this research is, for a certain class of objects, to determine whether it is possible to characterize those objects which have certain properties for pr ovable reasons. Surprisingly, it can be shown that in some cases this is not possible---which objects have the specified property depends on global features of the entire universe of sets. Another subject of this research is class forcing. Essentially, only one method is known for proving that statements are independent, namely, Cohen's forcing method. Recent work has shown that there is a good reason for this. In many cases, any independence result that can be obtained can be obtained by class forcing, in one sense of the term. However, this sort of class forcing is combinatorially far removed from the sort that is typically used to establish independence results. It is a goal to close this gap (or to explore why it is not possible to close it). A hope is that this will provide a means of reducing certain problems to tractable questions regarding forcing.

斯坦利建议继续他的工作集理论在两个领域，即无限组合学和类强迫。一组组合问题涉及表征哪些固定集可以包含封闭的无界子集在基数和GCH保持外模型。每一个静止子集的aleph-one有这个属性。以前的工作表明，在一般情况下，没有这样的一阶表征的子集的aleph-two。 另一方面，对于有限类的子集的继任者的aleph-omega，这样的表征是可能的。一些有趣的案件仍然悬而未决。关于分区和树，可以提出类似的特征化问题。另一个需要继续研究的领域是外部模型和阶级强迫的研究。尽管有一些技术条件，基本上只有一种方法是已知的构建外部模型，即科恩的强迫方法。到目前为止的工作已经表明，这不是一个偶然：在某种意义上，给予一个技术假设，每个外部模型都是一个类强制扩展。一个开放的问题是描述外部模型，在某种更强的意义上是类强制扩展。这项工作的一个重点是减少某些集合理论问题，以更具体的组合问题。 所有的数学陈述都可以转换成关于集合的陈述。 如果一个给定的陈述在数学上是可证明的，那么关于集合的相应陈述就可以从集合论的公理推导出来。 因此，从集合论的公理中翻译出来的既不可证明也不可反驳的陈述本身在数学上既不可证明也不可反驳。 这种声明被称为“独立的”。 在某些情况下，一个给定的数学对象可能具有某种性质，这是由于可证明的原因;在其他情况下，由于独立的原因。 本研究的一个主题是，对于某一类对象，确定是否可能表征那些由于合理的原因而具有某些性质的对象。 令人惊讶的是，可以证明，在某些情况下，这是不可能的-哪些对象具有指定的属性取决于整个集合论域的全局特征。 本研究的另一个主题是阶级压迫。 从本质上讲，只有一种方法是已知的证明陈述是独立的，即科恩的强迫方法。 最近的研究表明，这是有充分理由的。 在许多情况下，任何可以获得的独立性结果都可以通过类强制获得，在术语的某种意义上。 然而，这种类别强制在组合上与通常用于建立独立性结果的排序相去甚远。 我们的目标是缩小这一差距（或探讨为什么不可能缩小这一差距）。一个希望是，这将提供一种手段，减少某些问题，以易于处理的问题，有关强迫。