Mathematical Sciences: Inevitability in Model Theory

9403701 Laskowski博士将继续他在数理逻辑的一个分支——模型论领域的研究。他希望获得更多关于在不具有维序性质（DOP）的超稳定理论中使一种类型不可避免的可能机制的信息。拉斯科夫斯基希望完成他对那些有两个非同构模型的可数理论的分类，这些模型可以通过具有可数链条件的强制来强制同构。最后，他将继续研究两种构建小型超稳定理论模型的新方法之间的内在不对称性。模型理论（作为数理逻辑的一个分支）的大部分领域都关注于开发具有某些性质的代数结构的构建方法。是否存在一个结构的元素满足无限属性集（即实现类型）的问题是很容易理解的：这样的结构存在当且仅当，对于类型的每个有限子集，有一个元素满足这些属性中的每一个。然而，询问给定类型是否不可避免是一个更微妙的问题，即是否每个结构都包含实现该类型的元素。众所周知，如果不对所考虑的结构类别增加额外的限制，就不可能有令人满意的标准来确定哪些类型是不可避免的。拉斯科夫斯基将继续他的研究，在代数结构上必须加上什么额外的约束，以产生对必然性的自然表征。＊＊＊

9403701 Laskowski Dr. Michael Laskowski will continue his research in the field of model theory, a branch of mathematical logic. He expects to obtain more information as to the possible mechanisms making a type inevitable in the setting of superstable theories that do not have the Dimensional Order Property (DOP). Laskowski expects to complete his classification of those countable theories for which there are two non-isomorphic models that can be forced isomorphic by a forcing with the countable chain condition. Finally, he will continue his research into an inherent asymmetry between two new methods of constructing models of small, superstable theories. Most of the field of model theory (as a branch of mathematical logic) is concerned with developing methods for building algebraic structures with certain properties. The problem of whether there is a structure having an element satisfying an infinite set of properties (i.e., realizing a type) is well understood: such a structure exists if and only if, for every finite subset of the type, there is an element satisfying each of these properties. However, it is a much more subtle question to ask whether a given type is inevitable, i.e., whether EVERY structure contains an element realizing the type. Without adding additional restrictions to the class of structures under consideration, it is known that there can be no satisfactory criterion for which types are inevitable. Laskowski will continue his research on what additional constraints must be placed on the algebraic structure to yield a natural characterization of inevitability. ***