Mathematical Sciences: Computability and Mathematical Definability

9500878 Slaman Slaman打算研究可计算性和数学可定义性。这项研究将有助于发展对与相对可定义性相关的度理论结构的完整理解，例如图灵度D的全局结构和递归可枚举集R的图灵度R的全局结构。这种理解所缺少的要素包括确定R的普遍存在论是否可决定，表征R或D上的关系何时在该结构中可定义，计算这些结构的自同构群。其次，Slaman计划对二阶算法的证明理论理解做出贡献。在这种情况下，我们考虑形式公理系统，它假定宇宙（集合或实数）在可定义运算下是封闭的。例如，考虑一组公理，说明实数在算术可定义性的相对计算下是封闭的。在这种情况下，Slaman计划系统地研究连续统的组合集合论性质。他对守恒问题特别感兴趣：什么时候连续统的闭包性质对一阶数论有重要的影响？可定义性的代数结构可以很好地用图灵度来表示，图灵度是整数集合的一种分类，其中两个可以相互计算的集合被认为是等价的。Slaman将在几种情况下研究图灵学位，以解决有关其关键特征的基本问题。非有效手段的必要性，如不可计算集合的存在性，也可以从理论上证明；这激发了Slaman对连续统（即与所有实数相等的无限集）的组合集论性质进行系统研究的计划。他对将连续统的集合论与其他重要集合联系起来的问题特别感兴趣。什么时候连续统的某些性质对整数的初等性质有影响？* * *

9500878 Slaman Slaman intends to study computability and mathematical definability. This research will contribute to developing a complete understanding of the degree-theoretic structures associated with relative definability, such as the global structures of the Turing degrees D and the Turing degrees of the recursively enumerable sets R. Missing ingredients necessary to this understanding include determining whether the universal-existential theory of R is decidable, characterizing when a relation on R or D is definable within that structure, and calculating the automorphism groups of these structures. Secondly, Slaman plans contributions to the proof-theoretic understanding of second-order arithmetic. In that context, one considers formal axiom systems which postulate that the universe (of sets or of reals) is closed under definable operations. For example, one considers the collection of axioms that states that the reals are closed under relative computation of arithmetic definability. Slaman plans systematically to study the combinatorial set-theoretic properties of the continuum in this context. He is particularly interested in conservation questions: When do closure properties of the continuum have nontrivial consequences for first-order number theory? The algebraic structure of definability is well represented by the Turing degrees, a classification of sets of integers in which two sets that can be computed from each other are considered equivalent. Slaman will study the Turing degrees in several settings, to address fundamental questions about their critical features. The necessity of noneffective means, such as the existence of sets which are not computable, can also be demonstrated proof-theoretically; this motivates Slaman's program for the systematic investigation of the combinatorial set-theoretic properties of the continuum (i.e., of an infinite set of size equal to that of all real numbers). He is particularl y interested in questions which relate the set theory of the continuum to other important sets. When do certain properties of the continuum have bearing on elementary properties of the integers? ***