Basis Problems in Set Theory

集合论的基础问题

• 批准号: RGPIN-2014-04184
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=610882
• 关键词: Basis; Problems; Set; Theory

We propose to continue the project of developing a set-theoretic coarse classification theory for non-separable mathematical structures with an eye towards specific applications to surrounding areas of mathematics. This time we would like also to examine its application to set theory itself and in particular to the continuum problem. Many coarse classification problems at this level concern structures that can be put to live on the set of countable ordinals, a set that plays the crucial role in any formulation of the continuum problem so it is not surprising that there should be some connection here at all. As it turns out, any set-theoretic coarse structure theory developed so far goes in the opposite to the original Cantor's Continuum Hypothesis. In fact, these theories tend to give quite specific values to the cardinality of the continuum and other cardinal invariants of the continuum, and so we would like to examinee this phenomena from a general perspective. Some specific coarse classification problems that we propose to work on is the basis problem for compact Hausdorff spaces and the separable quotient problem for general Banach spaces. In particular, we would like to find the exact conditional structure that is present in any sufficiently long weakly null sequence. It turns out that the coarse set-theoretic classification theory goes well with the classical approaches of this kind such as, for example, the Ramsey classification theory and the Tukey classification theory. So part of our project will be devoted towards examining these connections as well. In fact, we have recently discovered that there is a quite close relationship between Tukey classification theory of ultrafilters on the set of natural numbers and the Ramsey theory of countable structures. This is leading us to new problems of Ramsey theory that are of independent interests and are likely to have further applications. So, a part of our efforts during the project will be devoted towards developing this kind of Ramsey theory as well. This will be a natural continuation of our already old project towards developing Ramsey theory this is applicable not only to set theory but also to other surrounding areas of mathematics such as, for example, the topological dynamics of groups of automorphisms of ultrahomogeneous structures.

我们建议继续开发不可分数学结构的集合论粗分类理论的项目 着眼于数学领域的具体应用。这一次，我们还想研究它在集合论本身，特别是连续统问题上的应用。这一级别的许多粗略分类问题都涉及可以放在可数序数集上的结构，可数序数集在任何连续统问题的表述中都扮演着关键角色，因此这里应该有一些联系也就不足为奇了。事实证明，到目前为止发展起来的任何集合论粗结构理论都与最初的康托的连续统假设背道而驰。事实上，这些理论倾向于给连续统的基数和连续统的其他基数不变量赋予相当具体的值，所以我们想从一般的角度来审视这一现象。我们提出的一些具体的粗分类问题是紧Hausdorff空间的基问题和一般Banach空间的可分离商问题。特别是，我们希望找到在任何足够长的弱零序列中存在的确切的条件结构。结果表明，粗集理论分类理论与Ramsey分类理论和Tukey分类理论等经典分类方法具有很好的一致性。因此，我们项目的一部分也将致力于研究这些联系。事实上，我们最近发现，自然数集上的超滤子的Tukey分类理论与可数结构的Ramsey理论之间存在着相当密切的关系。这将我们引向拉姆齐理论的新问题，这些问题具有独立的利益，并可能有进一步的应用。因此，我们在项目期间的部分努力也将致力于发展这种拉姆齐理论。这将是我们发展Ramsey理论的旧项目的自然延续这不仅适用于集合论，而且还适用于数学的其他周围领域，例如，超均匀结构的自同构群的拓扑动力学。