Basis Problems in Set Theory

我建议研究关于给定结构A的以下类型的问题，我将其称为‘A的展开问题’：给定一个正整数k，是否有一个有限的附加关系和函数列表可以放在A上，以便我们可以捕捉到A上的任何k元关系，即对于A上的每个k元关系R，都有A的一个‘大’子结构X，使得R对X的限制可以在仅限于X的展开结构中简单地定义？在不同的语境中，“大”的解释是不同的，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A类中求解这类问题，其中‘大’被解释为包含A的任意有限生成的子结构的同构副本，扩展问题等价于Ramsey度问题，并且这又等价于A的自同构群的泛极小流作为该群在A的所有扩展的特定可度量紧致空间上的作用的表示问题。解决其他结构的扩展问题，例如实数域，与集合论的其他领域和其他潜在的数学应用有关。值得一提的是，这类展开问题的研究是我过去投入大量时间研究集合论和Ramsey理论基础问题的自然延伸，但新的焦点给我们带来了到目前为止还没有完全认识到的优势。其中一个新特点是使用大基数来寻找各种展开问题的解决方案，将我们带入传统上被认为相距甚远的集合论部分之间的意想不到的联系。对扩展问题的新关注带来的另一个特征也是潜在的数学应用。我计划花一些时间来寻找对这一现象的适当解释，以将前面提到的扩展问题与拓扑动力学问题的联系相匹配。作为一个测试案例，我们以实数域为例，当“大”被解释为“本身非空且稠密”的实数集时，即有理数的拓扑副本。我们知道，在这种情况下，REAL的良好排序可能是一种额外的二元关系，它将解决扩展问题，使我们在取得进一步进展方面具有相当大的优势。更准确地说，我们得到了一个使用大基数的结果，它表明，实数的良好排序解决了可以添加到实数领域的二元关系的扩展问题。在这种情况下，当我们将“大”解释为“不可数”时，我们得到了与关于连续统的基数的问题的有趣的联系，我也打算研究这个问题。

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.