RUI: Outer Models and Forcing

Stanley建议在两个领域继续他在集合论方面的工作，即无穷组合学和类强迫。一组组合问题涉及到在基数和GCH保持外模型中哪些平稳集可以包含闭无界子集的特征。1的每个平稳子集都有这个性质。先前的工作表明，一般来说，对于alpha - 2的子集没有这样的一阶表征。另一方面，对于后继的子集的一个有限类，这样的刻画是可能的。一些有趣的案件仍未结案。关于分区和树，也可以提出类似的特征问题。另一个需要继续工作的领域是对外部模式和阶级强迫的研究。尽管有一些技术上的限制，但基本上只有一种方法已知用于构建外部模型，即科恩强迫法。到目前为止的工作表明，这不是偶然的：在某种意义上，每个外部模型都是一个类强制扩展。一个有待解决的问题是，如何描述在某种更强意义上是类强制扩展的外部模型。这项工作的一个重点是将某些集合理论问题简化为更具体的组合问题。所有的数学陈述都可以转化为关于集合的陈述。如果一个给定的命题在数学上是可证明的，那么相应的关于集合的命题就可以从集合论的公理中推导出来。因此，从集合论的公理中转换出来的既不能证明也不能反驳的命题，在数学上也不能证明也不能反驳。这样的陈述被称为“独立的”。在某些情况下，一个给定的数学对象可能由于可证明的原因而具有某种性质；在其他情况下，出于独立的原因。这项研究的一个主题是，对于某一类物体，确定是否有可能表征那些具有某些特性的物体，这些特性是出于合理的原因。令人惊讶的是，可以证明在某些情况下这是不可能的——哪个对象具有指定的属性取决于整个集合的全局特征。这项研究的另一个主题是阶级强迫。从本质上讲，只有一种方法可以证明陈述是独立的，即科恩强迫法。最近的研究表明，这是有充分理由的。在许多情况下，可以获得的任何独立性结果都可以通过类强制获得，这是该术语的一种意义。然而，这种类型的类强制在组合上与通常用于建立独立性结果的类型相去甚远。我们的目标是缩小这个差距（或者探索为什么不可能缩小差距）。希望这将提供一种手段，将某些问题简化为有关强迫的可处理问题。

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.