不连续性、极限可计算性和跳跃算子的水平
Levels of discontinuity, limit-computability, and jump operators
摘要：
我们开发了一个通用的跳跃算子理论，其目的是提供一个抽象的概念“极限可计算性”的代表空间。跳跃算子还提供了一个框架，具有很强的分类风味，用于研究所表示空间上的函数和集合层次的不连续性程度。我们将提供一个彻底的调查在这个框架内的层次结构的$Δ^0_2$ -可可测函数之间的任意可数为基础的$T_0$ -空间，它捕获的概念计算与顺序的思维变化的界限。我们的抽象方法不仅提出了新的问题，但也揭示了新的光在以前的结果。例如，我们引入了一个概念，“高阶”描述集理论的对象，我们概括了最近表征的计算理论概念的“低”的伴随函子，我们表明，我们的框架包括序量化的非建设性希尔伯特有限基础定理。
英文摘要：
We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of $Δ^0_2$-measurable functions between arbitrary countably based $T_0$-spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of "higher order" descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of "lowness" in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert's finite basis theorem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
