Large Cardinals and the Methodology of Mathematics

首席研究员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较大的基数假设，其中一些假设是必要的。该资助项目的一个研究方向是调查具有相同属性的对象类别，即所谓的终端类别。一个典型的结果表明Ramsey超滤子是一个终端类：它们大致具有滤子在Rudin-Keisler等价下不变的所有性质。正在调查的开放问题包括找到更多这样的终端类，更重要的是，量化这些类在整个数学中的扩散。另一个研究方向是建立连续统的基本不变量行为为最优的模型。一个典型的结果是，米勒模型是增加主导数d的最优方法：粗略地说，在这个模型中，所有一致小于d的投影定义不变量都小于d。对偶结果表明存在一个边界数较小的最优Pmax模型。开放问题涉及找到与之相关的规范模型的进一步基数不变量。更具挑战性的是对上面提到的对偶概念的研究，以及对单可定义偏序的强迫方法的极限的研究。集合理论家长期以来一直在研究一些额外的数学公理，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具体问题无关，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语法形式来选择回答问题的最佳方法。通常，这些信息可以用来发现看似复杂问题的核心。有三个例子。作为第一个例子，二十年来人们已经知道，从数学分析的角度来看，具有简单定义的实数集表现良好。其次，PI在数学实践中确定了几类“终端”对象：此类中的所有对象具有相同的属性。这样的类具有重要的方法论意义，PI计划将更多的类隔离出来。在另一个发展中，PI发现，对于某些数学上重要的理论类别，存在一个最佳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。总的来说，资助的项目旨在表明，像大基数公理这样看似深奥的假设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互动。

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.