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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=689829
• 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.

我建议研究关于给定结构A的以下类型的问题，我称之为“A的扩展问题”：给定一个正整数k，有一个有限的附加关系和函数列表，可以放在A上，这样我们就可以捕获A上的任何k元关系，对于每个k-元关系R在A上有一个“大”子结构X的A，使限制的R在X上是简单地定义在扩展的结构限制到X？在不同的上下文中，“大”的解释不同，但我们通常会对“大”的最佳限制感兴趣。我们已经看到，在可数超齐次结构类A中解决这类问题，其中“大”被解释为包含A的任意非齐次生成子结构的同构副本，膨胀问题相当于拉姆齐问题度问题，这又等价于A的自同构群的泛最小流的表示问题，作为这个群在一个特定的A的所有扩张的可度量化紧空间。解决其他结构的扩展问题，例如真实的数域，与集合论的其他领域有关，也与其他潜在的数学应用有关。应该提到的是，这种扩展问题的研究是一个自然延伸的研究集理论和拉姆齐理论的基础问题，我已经投入了相当多的时间在过去，但新的重点给我们带来的优势，尚未完全认识到迄今为止。其中一个新的特点是使用大基数在寻找解决方案的各种扩展问题，使我们的一个意想不到的连接部分集理论，一直被认为是相距甚远。另一个特点，新的重点扩大问题带来的是，以及潜在的数学应用。我计划花一些时间来找到一个适当的解释，这个现象可以匹配前面提到的膨胀问题与拓扑动力学问题的联系。作为一个测试案例，我们采取了真实的数字的领域时，“大”被解释为“非空和密集本身”的一组真实的数字，即，有理数的拓扑副本我们知道，在这种情况下，实数的良序可能是一个额外的二元关系，它将解决扩展问题，为我们进一步的进展提供相当大的优势。更确切地说，我们有一个结果，它表明使用大基数，一个良好的秩序的实数解决了扩展问题的二元关系，可以添加到该领域的真实的号码。当在这种情况下，我们解释'大'为'不可数'，我们得到了一个有趣的问题，关于基数的连续，我也打算检查。