Large Cardinals and the Methodology of Mathematics

主要研究者研究大基数对数学方法的影响。下面所述的所有结果都使用了大基数假设，其中一些假设是必要的。资助项目的一个研究方向是研究具有相同属性的对象类，即所谓的终端类。一个典型的结果表明Ramsey超滤子是一个终端类：粗略地说，它们共享所有在Rudin-Keisler滤子等价下不变的性质。正在调查的开放性问题涉及进一步寻找这样的终端类，更重要的是，量化整个数学中这样的类的扩散。另一个研究方向是构建模型，其中连续体的基数不变量的行为是最优的。一个典型的结果是，米勒模型是增加控制数d的最佳方法：粗略地说，所有一致小于d的射影定义的不变量在这个模型中都小于d。对偶结果表明，存在一个最优的Pmax模型，其中边界数很小。开放的问题涉及到寻找进一步的基数不变量，其中有与他们相关的典型模型。 更有挑战性的是研究上面提到的对偶概念，以及研究用可简单定义的偏序强迫方法的局限性。集合论学家长期以来一直在研究数学中的某些附加公理，称为大基数公理。虽然它们在很大程度上与解决大多数传统数学领域的特定问题无关，但它们确实对所使用的方法有很大的影响。通常，大基数公理允许数学家仅通过考虑问题的句法形式来选择回答问题的最佳方法。这些信息常常有助于发现一个看似复杂的问题的核心。 有三个例子。 作为第一个例子，二十年前人们就已经知道，从数学分析的角度来看，具有简单定义的实数集表现良好。第二，PI在数学实践中识别了几类“终端”对象：此类对象中的所有对象都具有相同的属性。这样的类具有很大的方法论意义，PI计划分离更多的类。在另一个发展中，PI发现，对于某些数学上重要的理论类别，存在一种最优方法来回答理论是否不包含矛盾的问题。同样，PI继续进一步隔离此类类别。一般来说，资助的项目旨在表明，这些看似深奥的假设，如大基数公理，对数学实践有直接的影响，从而促进逻辑，集合论和其他数学分支之间的相互作用。

The principal investigator studies the impact of large cardinals on the methodology of mathematics. All results stated below use large cardinal assumptions, and some such assumptions are necessary. One line of research for the funded project is the investigation of classes of objects with the same properties, so-called terminal classes. A typical result states that the Ramsey ultrafilters are a terminal class: roughly, they share all properties invariant under the Rudin-Keisler equivalence of filters. Open questions under investigation involve finding further such terminal classes, and more importantly, the quantification of theproliferation of such classes throughout mathematics. Another line of research pursued is the construction of models in which the behavior of cardinal invariants of the continuum is optimal. A typical result is that the Miller model is the optimal way of increasing the dominating number d: roughly, all projectively defined invariants which are consistently less than d, are less than d in this model. The dual result states that there is an optimal Pmax model in which the bounding number is small. Open problems involve finding further cardinal invariants which have canonical models associated with them. More challenging is the investigation of the notion of duality mentioned above, and the investigation of the limits of the method of forcing with simply definable partial orders.Set theorists have for a long time studied certain additional axioms for mathematics, called large cardinal axioms. While they are largely irrelevant for solving specific problems in most traditional fields of mathematics, they do have a strong influence on the methodology used. Typically the large cardinal axioms allow the mathematician to select an optimal approach to answering a question only by considering the syntactical form of the question. Frequently this information can serve to discover the core of a seemingly complex problem. Three examples are in order. As the first example, it has been known for twenty years that sets of reals with simple definitions are well behaved from the point of view of mathematical analysis. Second, the PI has identified several classes of objects in mathematical practice that are "terminal": all objects in such a class have the same properties. Such classes have great methodological significance, and the PI plans to isolate more of them. In still another development, the PI found that for certain mathematically important classes of theories, there is an optimal approach to answering the question of whether the theories contain no contradictions. Again, the PI continues to isolate further such classes. Generally, the funded project serves to show that such seemingly esoteric hypotheses as large cardinal axioms have direct impact on mathematical practice, thus promoting the interaction between logic, set theory and other branches of mathematics.