Basis Problems in Set Theory

集合论的基础问题

• 批准号: RGPIN-2019-04311
• 负责人: Todorcevic, Stevo
• 金额: $ 3.64万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713911
• 关键词: Basis; Problems; Set; Theory

I propose to study the following type of a problem about a given structure A which I will call 'The Expansion Problem for A': Given a positive integer k, is there is a finite list of additional relations and functions that one can put on A so that we can capture any k-ary relation on A in the sense that for every k-ary relation R on A there is a 'large' substructure X of A so that the restriction of R on X is simply definable in the expanded structure restricted to X? In different contexts the 'large' is interpreted differently but we shall be typically interested in the optimal restriction on 'large'. We have seen that solving this type of problems in the class of countable ultra homogeneous structures A where 'large' is interpreted as containing isomorphic copy of an arbitrary finitely generated substructure of A, the expansion problem is equivalent to the Ramsey-degree problem and this is in turn equivalent to the representation problem of the universal minimal flow of the automorphism group of A as an action of this group on a particular metrizable compact space of all expansions of A. Solving the expansion problem for other structures such as for example the field of real numbers is related to other areas of set theory and to other potential mathematical applications. It should be mentioned that the study of expansion problems of this sort is a natural extension of the study of set-theoretic and Ramsey-theoretic basis problem that I have invested considerable time in the past but the new focus brings us advantages that have not fully recognized so far. One of the new feature is is the use of large cardinals in finding solutions to various expansion problems bringing us to an unexpected connections between parts of set theory that have been traditionally considered as far apart. Another feature which the new focus on expansions problem brings are as well the potential mathematical applications. I plan to invest some time in finding an appropriate explanation of this phenomenon that could match the aforementioned connection of the expansion problem with a problem from topological dynamics. As a test case we take the field of real number when 'large' is interpreted as 'nonempty and dense in itself' set of real numbers i.e., a topological copy of the rationals. We know that in this case a well-ordering of the reals could be an additional binary relation that would solve the expansion problem giving us considerable advantage towards further progress. More precisely, we have a result which shows using large cardinals that a well-ordering of the reals solves the expansion problems for binary relations that could be added to the field of real numbers. When in this context we interpret 'large' as 'uncountable' we get an interesting connection with the problem about the cardinality of the continuum which I also intend to examine.

我建议学习以下类型的一个给定的结构问题我将称之为“扩张问题”:给定一个正整数k,是有一个有限的附加的关系和功能列表可以穿上这样我们就可以捕获任何k-ary关系上,每k-ary关系R上有一个“大”的子结构X X R的限制在仅仅是可确定的X扩展结构的限制?在不同的上下文中，“大”有不同的解释，但我们通常感兴趣的是对“大”的最优限制。我们已经看到，在可数超齐次结构A中解决这类问题，其中“大”被解释为包含A的任意有限生成子结构的同构副本，展开问题等价于ramsey度问题而这又等价于A的自同构群的普遍最小流的表示问题作为这个群在A的所有展开的一个特定的可度量紧化空间上的作用，解决其他结构的展开问题例如实数域与集合论的其他领域有关