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

这个建议的主题是无限组合，这意味着无限数的组合。这些数字可以用来为持续无限多步骤的过程建模，从而了解这些过程的本质。这里的研究本质上是基础性的。它与应用程序的联系是，无限过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，包括我们在日常生活中使用的物理计算机，在很大程度上是基于有限组合的。无限基数分为两组，正则和奇异。正则基数的组合比奇异基数的组合更为人所知。这里我们研究单一枢机的直接继承者。该建议所涉及的具体主题是有一个单一基数的继承者的可能性，在这个基数上既有强制公理又有反射原则。这将与常规枢机主教继任者可能取得的成就形成鲜明对比。我们讨论了基数r0，即Rado的陈述成立的第一个无限基数，是否可能是一个奇异基数的后继。本研究的背景是建立在集合论的一个中心问题上，奇异基数假设SCH和理解奇异基数后继者的组合性质的相关问题。这些问题与希尔伯特清单上的第一个问题直接相关，这个问题已经有一个多世纪的历史了。这个提议带来了一种全新的技术来解决这个问题。本文的研究假设和目标是利用公理SSF的一个模型，通过Radin强迫得到一般扩展中奇异的后继。我们期望从这个项目中得到的其他结果是关于奇点后继的组合结果，我们希望在我们的模型确实给出0是一个奇点后继的情况下发现这些结果，但在它没有给出的情况下，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，然后是更广泛的数理逻辑学社区。随着结果的确定，我们期望在数理逻辑以外的领域应用，如巴拿赫空间和测量代数。PI在将集合论的进展与源自其他数学领域的问题联系起来方面有着相当丰富的经验。在英国，高端集合理论在布里斯托尔大学和东安格利亚大学的逻辑小组中都有。否则，不幸的是，在全国范围内，它的代表性不足。在其他欧洲国家，这是一个非常活跃的领域，包括奥地利、法国、德国和波兰，以及美国、以色列、加拿大，最近通过相当大的国家投资，澳大利亚也是如此。该领域已经得到了欧洲科学基金会（European Science Foundation）为研究网络INFTY颁发的一项大型欧洲赠款的认可。我们希望这项研究的结果能够引起大量国际顶尖研究人员的高度兴趣。

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.