Mathematical Sciences: Computability and Mathematical Definability

小行星9500878 斯拉曼打算研究可计算性和数学可定义性。 这项研究将有助于全面了解 与相对可定义性相关的度理论结构，例如图灵度D和图灵 递归可重集R的度。 缺失成分 这一理解所必需的包括确定 R的普适存在论是可判定的，当 R或D上的关系在该结构内是可定义的，并计算这些结构的自同构群。 第二、 Slaman计划对证明理论的理解做出贡献， 二阶算术 在这种情况下，我们考虑形式公理 假设宇宙（集合或实数）是 在可定义的操作下关闭。 例如，人们认为， 公理的集合，说明实数在 算术可定义性的相对计算。 斯拉曼计划 系统地研究组合集合论性质的 在这种情况下，连续。 他特别感兴趣的是 守恒问题：连续统的封闭性质 对一阶数论有重要影响吗 可定义性的代数结构可以用图灵度很好地表示，图灵度是一种整数集合的分类，其中两个集合 可以从彼此计算出的值被认为是等效的。 斯拉曼 我将在几种情况下研究图灵度，以解决基本的 关于其关键特征的问题。 无效的必要性 手段，如存在不可计算的集合，也可以是 从理论上证明了这一点;这激发了Slaman的计划， 的组合集合论性质的系统研究 连续体（即，一个无限的集合的大小等于所有真实的 数字）。 他特别感兴趣的问题， 连续统理论到其他重要集合。 当某些属性 和整数的基本性质有关系吗 ***

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? ***