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Combinatorial set theory at the successor of a singular cardinal: a marriage of a forcing axiom and a reflection principle

奇异基数后继的组合集合论：强制公理与反射原理的结合

基本信息

批准号：
EP/I00498X/1
负责人：
Mirna Dzamonja
金额：
$ 37.46万
依托单位：
University of East Anglia
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI00498X%2F1
关键词：
Combinatorial set theory successor singular

项目摘要

The subject of this proposal lies within infinite combinatorics, which means combinatorics of inifinite numbers. Such numbers can be used to model processes that last infinitely many steps and therefore to understand the nature of such processes. The research here is fundamental in nature. Its connection with applications is that infinite processes are expected to have a role in building future generations of artifical intelligence and computing equipment. The computing capabilities known now, including the physical computers that we use in everyday life, are based heavily on finite combinatorics. Infinite cardinal numbers come in two groups, regular and singular. Much more is known about the combonatorics of regular cardinals than the combinatorics of the singular ones. Here we study immediate successors of singular cardinals.The specific subject that the proposal is concerned with is the possibility of having a successor of a singular cardinal on which there is both a forcing axiom and reflection principle. This would stand in sharp contrast with what is possible to achieve at successors of regular cardinals. We address the question if it is possible that the cardinal r0, the first infinite cardinal at which Rado's statement holds, can be the successor of a singular cardinal. The background for this research builds on a central question in set theory, the singular cardinal hypothesis SCH and the related question of understanding the combinatorial nature of successors of singular cardinals. The questions are directly connected to the first problem on the Hilbert's list, now over a century old. The proposal brings an entirely novel technology by which to attack the problem.The research hypothesis and objectives are to use a model of the axiom SSF to get r0 the successor of a singular in a generic extension by Radin forcing. Other outcomes we expect from the project are combinatorial results about sucessors of singulars, and we expect to discover these in the case that our model indeed does give r0 is sucessor of a singular, but also in the case that it does not, because in the latter case we will have to understand why such a combination is not possible.The academic beneficiaries of this project are first of all the set-theoretic community, and then more widely the community of mathematical logicians. As the results become settled we expect applications in fields outside of mathematical logic, such as Banach spaces and measure algebras. The PI has a considerable experience of being able to connect advances in set theory with problems stemming from other areas of mathematics.In the UK, high-end set theory is present at Bristol and in the logic group at UEA. Otherwise, it is unfortunately underrepresented nationwide. It is a very active area in other European countries, including Austria, France, Germany and Poland, and in the United States, Israel, Canada and since recently through a considerable national investement, also Australia. The area has been recognised by the award of a large European grant for research networking INFTY, awarded by the European Science Foundation. We expect the results of this research to be of high interest to a large number of top researchers internationally.

这一建议的主题在于无限组合学，这意味着组合的无穷数。这些数字可以用来模拟持续无限多个步骤的过程，从而理解这些过程的本质。这里的研究在本质上是基础性的。它与应用的联系是，无限进程有望在构建未来几代人工智能和计算设备中发挥作用。现在已知的计算能力，包括我们日常生活中使用的物理计算机，都是基于有限组合学的。无限基数分为两组，正则基数和奇异基数。关于正则基数的组合学比奇异基数的组合学知道得多。在这里我们研究奇异基数的直接后继者。该建议所关心的具体问题是有一个奇异基数的后继者的可能性，在这个后继者上既有一个强制公理又有一个反射原理。这将与常规红衣主教的继任者可能实现的目标形成鲜明对比。我们解决的问题，如果它是可能的，基数r0，第一无限基数在Rado的声明举行，可以是一个单一的基数的继任者。本研究的背景建立在一个中心问题，在集合论，奇异基数假设SCH和相关问题的理解组合性质的后代奇异基数。这些问题直接与希尔伯特列表上的第一个问题有关，这个问题已经有世纪的历史了。该方案提出了一种全新的技术来解决这一问题，其研究假设和目标是利用公理SSF的模型，通过Radin强迫使r0成为类属扩张中奇异的后继。我们期望从这个项目中得到的其他结果是关于奇点后继者的组合结果，我们希望在我们的模型确实给出了r0是奇点后继者的情况下发现这些结果，但也希望在它没有给出的情况下发现这些结果，因为在后一种情况下，我们必须理解为什么这样的组合是不可能的。这个项目的学术受益者首先是集合论社区，以及更广泛的数学逻辑学家群体。随着结果得到解决，我们预计在数学逻辑领域以外的应用，如Banach空间和测度代数。PI有相当丰富的经验，能够连接在集合论的进步与数学的其他领域产生的问题。在英国，高端集合论是目前在布里斯托和逻辑组在UEA。除此之外，它在全国的代表性不足。这是一个非常活跃的领域，在其他欧洲国家，包括奥地利，法国，德国和波兰，并在美国，以色列，加拿大和最近通过一个相当大的国家投资，也澳大利亚。该地区已被欧洲科学基金会授予的欧洲大型研究网络INFTY奖所认可。我们预计这项研究的结果将引起国际上大量顶尖研究人员的高度兴趣。