Mathematical Sciences: Degree Structures and Forcing

本项目涉及集合理论和递归理论，重点研究不同的度结构和强迫在度理论问题中的应用。研究者将继续对递归可枚举度理论中的优先级论证进行分析，并对这些论证的理论证明强度进行研究。Groszek和Theodore a . Slaman（芝加哥大学）一直在为所有级别的优先级论证开发一种统一的表示，并将这些论证与皮亚诺算法片段的标准证明理论层次联系起来。除了继续这一发展，Groszek将研究关于递归可枚举图灵度的偏序的具体陈述，考虑它们的证明理论强度是否以及如何反映这一分析。Groszek将以不同的方式研究一些长期存在的关于可建构性程度的开放性问题。最后，她将考虑哪些偏阶可以嵌入图灵度的问题（假设CH失效），这是关于图灵度全局结构的剩余开放问题之一。在这个项目要解决的问题中，突出的是一些与理论可计算性有关的问题。它们存在于所谓的递归理论中，递归理论处理的是不受时间或空间限制的可计算性模型。虽然这些问题的答案有能力阐明实际问题，但只有当答案是否定的时候，它们才是真正实际的，因为即使在不限制可用资源的情况下，也不能计算某些东西，这确实是一个非常有力的说法。可计算性理论的精细结构有时与实际计算更相关，并且还将考虑其各个方面。该项目的主要目的是将可计算性的程度等同于处理算术逻辑的经过充分研究的层次结构中的特定级别。

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.