Logic, Sets, Categories, and Applications

逻辑、集合、类别和应用

• 批准号: 0653696
• 负责人: Andreas Blass
• 金额: $ 36.51万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653696&HistoricalAwards=false
• 关键词: Logic; Sets; Categories; Applications

The proposer continues his research on applications of set-theoretic methods to abelian group theory, with particular emphasis on products of infinite cyclic groups and related groups. In addition to applying set-theoretic ideas to this branch of algebra, he is looking for new set-theoretic concepts and results motivated by algebraic problems. He is exploring the connection between universal algebra (in Lawvere's category-theoretic formulation), the theory of classifying topoi, and the theory of unification (which plays a prominent role in some areas of computer science). The connection is already understood in the case of absolutely free algebras, but further exploration is needed in the more general case of algebras subject to nontrivial identities. Unification in the presence of identities is in general a troublesome topic, and it is hoped that the connection with other, better understood areas can clarify it. The proposer is also studying a topos-theoretic approach to extending the notion of Borel functions to higher types. Such an extension is useful for the theory of cardinal characteristics of the continuum.Borel functions at the base type (real numbers) already play a major role in this theory, but so does an operation of sequential composition that necessarily leads to higher types. The proposer is also studying several questions at the border between constructive logic, category theory, and game semantics. This research includes an attempt to amplify the connection between games and free (or co-free) bicomplete categories, axiomatization of game-semantical validity, and a game-theoretic principle that may be fruitfully added to intuitionistic logic. Finally, the proposer is studying several questions in set theory, most of which are connected with ultrafilters and in particular with the equivalence relation of near coherence on ultrafilters. There are also some questions in infinite combinatorics growing out of considerations in abelian group theory. The project builds on the proposer's past experience in several areas of mathematics --- set theory, category theory (especially topos theory), abelian group theory, game semantics, constructive mathematics, and computer science --- in order to address problems bridging these areas. Almost all of the topics in the proposal connect two or more of these areas. In each case, it is reasonable to expect that ideas and styles of reasoning from one area can benefit other areas. Among the potential outcomes of this research are: (1) a better understanding of unification, which is an essential ingredient of logic programming, (2) a clearer view of constructive mathematical reasoning as based on strategies in certain sorts of debates, and (3) applications of a context that resembles traditional set theory in many ways yet avoids the phenomena sometimes regarded as pathological.

作者继续研究集合论方法在交换群论中的应用，特别是无限循环群及其相关群的乘积。除了将集合论思想应用于这个代数分支，他还在寻找新的集合论概念和结果，这些概念和结果是由代数问题所驱动的。他正在探索普适代数(在Lawvere的范畴理论公式中)、拓扑分类理论和统一理论(在计算机科学的某些领域中扮演着重要角色)之间的联系。在绝对自由代数的情况下，这种联系已经被理解，但在更一般的代数服从非平凡恒等式的情况下，还需要进一步的探索。在有身份的情况下统一通常是一个麻烦的话题，希望与其他更好理解的领域的联系能够澄清这一问题。作者还研究了一种拓扑论方法，将Borel函数的概念推广到更高的类型。这种推广对于连续统的基数特征理论是有用的。基型(实数)上的Borel函数已经在这一理论中发挥了重要作用，但必然导致更高类型的序列合成的运算也是如此。作者还研究了构造性逻辑、范畴理论和博弈语义学之间的几个问题。这项研究包括试图放大游戏和自由(或共同自由)双完全范畴之间的联系，游戏语义有效性的公理化，以及可以有效地添加到直觉主义逻辑中的博弈论原理。最后，作者研究了集合论中的几个问题，这些问题大多与超滤器有关，特别是与超滤器上的近相干性等价关系有关。由于交换群论的考虑，无限组合学中也存在一些问题。该项目建立在建议者过去在几个数学领域的经验-集合论、范畴理论(特别是拓扑论)、阿贝尔群论、游戏语义学、建构主义数学和计算机科学-以解决连接这些领域的问题。提案中的几乎所有主题都与上述两个或两个以上领域有关。在每一种情况下，都有理由期待来自一个领域的想法和推理风格可以造福于其他领域。这项研究的潜在结果包括：(1)更好地理解了统一，这是逻辑编程的一个重要组成部分；(2)在某些类型的辩论中，基于策略的建构性数学推理有了更清晰的观点；(3)应用了在许多方面类似传统集合论的背景，但避免了有时被视为病态的现象。