Mathematical Sciences: Degree Structures and Forcing

这个项目涉及集合论和递归理论， 侧重于各种程度的结构和应用 强迫解决学位理论问题。 研究者将 继续分析《公约》中关于优先权的论点， 递归可重复度理论及其证明理论 这些论点的力量。 Groszek和西奥多A.斯拉曼大学 （芝加哥）一直在为所有人制定统一的演示文稿 优先级参数的级别，并将这些参数绑定到标准 皮亚诺算法片段的证明理论层次。 在 除了继续这一发展，Groszek将调查 关于递归的偏序的具体陈述 可验证的图灵度，考虑到他们的证明是否以及如何- 这一分析在理论上有一定的说服力。 换句话说， Groszek将致力于解决一些长期悬而未决的问题， 可建设性的程度。 最后，她会考虑 问题是哪些偏序可以嵌入到图灵 度（假设CH失败），剩余的开放之一 关于图灵度的全局结构的问题。 在本项目要解决的问题中， 是一个与理论上的可计算性有关的数字。 这两个前轮位于一 即所谓的递归理论，它研究的是 不受时间和空间限制的可计算性。 虽然答案 这些问题有能力阐明实际的 问题，只有当他们的答案是 否定的，因为这确实是一个非常强烈的声明， 即使没有限制， 为此目的提供的资源。 更精细的结构 可计算性理论有时与实际更相关 计算，以及它的各个方面也将被考虑。 该项目的一个主要目标是将可计算性的程度等同起来 在一个经过充分研究的层次结构中， 算术逻辑

This project involves both set theory and recursion theory, focusing on various degree structures and the applications of forcing to degree-theoretic questions. The investigator will continue to work on the analysis of priority arguments in the theory of recursively enumerable degrees and on the proof-theoretic strength of these arguments. Groszek and Theodore A. Slaman (Univ. of Chicago) have been developing a uniform presentation for all levels of priority argument and tying these arguments to a standard proof-theoretic hierarchy of fragments of Peano arithmetic. In addition to continuing this development, Groszek will investigate specific statements about the partial order of recursively enumerable Turing degrees, considering whether and how their proof- theoretic strength reflects this analysis. In a different vein, Groszek will work on some long-standing open questions regarding the degrees of constructibility. Finally, she will consider the question of which partial orders can be embedded into the Turing degrees (assuming the failure of CH), one of the remaining open questions about the global structure of the Turing degrees. Prominent among the questions to be addressed by this project are a number that bear on theoretical computability. They lie in what is known as recursion theory, which deals with a model of computability knowing no bounds on time or space. Although answers to such questions have the ability to illuminate practical questions, they are really practical only when their answers are negative, for it is a very strong statement indeed to say that something cannot be computed even when one puts no limits on resources available for the purpose. The finer structure of computability theory is sometimes more relevant to actual computations, and various aspects of that will also be considered. A major thrust of the project is to equate degrees of computability with specific levels in a well studied hierarchy dealing with the logic of arithmetic.