Large Cardinals and the Methodology of Mathematics

大基数和数学方法论

• 批准号: 0071437
• 负责人: Jindrich Zapletal
• 金额: $ 6.14万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071437&HistoricalAwards=false
• 关键词: Large; Cardinals; Methodology; Mathematics

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.

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大的基本假设，并且一些这样的假设是必要的。该资助项目的研究方向之一是调查具有相同属性的对象类别，即所谓的终端类别。典型的结果表明拉姆齐超滤器是一个终端类别：粗略地说，它们共享过滤器 Rudin-Keisler 等价下的所有不变属性。正在研究的开放问题涉及进一步寻找此类终端类，更重要的是，对此类类在数学中的扩散进行量化。所进行的另一项研究是构建模型，其中连续统的基数不变量的行为是最优的。典型的结果是，米勒模型是增加主导数 d 的最佳方式：粗略地说，所有投影定义的始终小于 d 的不变量在该模型中都小于 d。对偶结果表明存在一个边界数较小的最优 Pmax 模型。开放问题涉及寻找更多具有与之相关的规范模型的基数不变量。 更具挑战性的是对上述对偶性概念的研究，以及对具有简单可定义的偏序的强制方法的限制的研究。集合论学家长期以来研究了某些附加的数学公理，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，大基本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。这些信息通常可以帮助发现看似复杂问题的核心。 三个例子按顺序排列。 作为第一个例子，二十年来人们都知道，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：此类中的所有对象都具有相同的属性。此类类具有重大的方法论意义，PI 计划隔离更多此类类。在另一个发展中，PI 发现对于某些数学上重要的理论类别，有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。一般来说，资助的项目旨在表明大基公理等看似深奥的假设对数学实践有直接影响，从而促进逻辑、集合论和其他数学分支之间的相互作用。