Basis Problems in Set Theory

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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.