马丁猜想的结果

P. Lutz
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引用次数: 1

摘要

摘要马丁猜想是在强集合论假设下对图灵度上所有可定义函数的行为进行分类的一种尝试。粗略地说,每一个这样的函数要么最终是常数,要么最终等于恒等函数,要么最终等于图灵跳跃的一个超限迭代。它通常分为两部分:第一部分表明,每个函数要么最终是常数,要么最终高于恒等函数;第二部分表明,每个高于恒等函数的函数最终等于跳跃的超限迭代。如果这是真的,它将为图灵跃迁在可计算性理论中的独特作用提供解释,并排除图灵度上的许多类型的结构。在这篇论文中,我们将介绍一些我们用来证明马丁猜想的几个例子的工具。结果是这些工具和Martin猜想的结果对Martin猜想和一些相关的话题都有一些有趣的结果。我们引入的主要工具是一个完备集的基定理,它改进了grosszek和Slaman的一个定理。我们还引入了一个证明马丁猜想某些特殊情况的一般框架,它统一了一些先前存在的证明。我们将使用这些工具来证明关于Martin猜想的三个主要结果:它对超算术度上的回归函数成立(回答了Slaman和Steel的问题),第1部分对图灵度上的保序函数成立,第1部分对我们引入的一类函数成立,称为保测度函数。最后这个结果对马丁猜想的研究有几个有趣的结果。特别地,它证明了Martin猜想的第一部分等价于图灵度上超滤波器上的Rudin-Keisler阶的陈述。这为我们将讨论的Martin猜想的第1部分提出了几种可能的策略。我们用来证明这些结果的基定理在马丁猜想之外也有一些应用。我们将用它来证明与Sacks的问题有关的几个定理,该问题是关于在$\mathsf {ZFC}$中是否可以证明每个局部可数的大小的偏序连续序列嵌入到图灵度中。我们将证明,在$\mathsf {ZF}$的某个扩展中(它与$\mathsf {ZFC}$不兼容),这适用于高度为2的所有偏阶,但不适用于高度为3的所有偏阶。对于$\mathsf {ZF}$中的Borel偏阶和Borel嵌入,我们的证明也得到了一个类似的结果,这表明Sacks问题的Borel版本有一个否定的答案。我们将以一系列与马丁猜想相关的悬而未决的问题来结束本文,我们希望这些问题能激发进一步的研究。摘要由Patrick Lutz准备。电子邮件:pglutz@berkeley.edu
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Results on Martin’s Conjecture
Abstract Martin’s conjecture is an attempt to classify the behavior of all definable functions on the Turing degrees under strong set theoretic hypotheses. Very roughly it says that every such function is either eventually constant, eventually equal to the identity function or eventually equal to a transfinite iterate of the Turing jump. It is typically divided into two parts: the first part states that every function is either eventually constant or eventually above the identity function and the second part states that every function which is above the identity is eventually equal to a transfinite iterate of the jump. If true, it would provide an explanation for the unique role of the Turing jump in computability theory and rule out many types of constructions on the Turing degrees. In this thesis, we will introduce a few tools which we use to prove several cases of Martin’s conjecture. It turns out that both these tools and these results on Martin’s conjecture have some interesting consequences both for Martin’s conjecture and for a few related topics. The main tool that we introduce is a basis theorem for perfect sets, improving a theorem due to Groszek and Slaman. We also introduce a general framework for proving certain special cases of Martin’s conjecture which unifies a few pre-existing proofs. We will use these tools to prove three main results about Martin’s conjecture: that it holds for regressive functions on the hyperarithmetic degrees (answering a question of Slaman and Steel), that part 1 holds for order preserving functions on the Turing degrees, and that part 1 holds for a class of functions that we introduce, called measure preserving functions. This last result has several interesting consequences for the study of Martin’s conjecture. In particular, it shows that part 1 of Martin’s conjecture is equivalent to a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. This suggests several possible strategies for working on part 1 of Martin’s conjecture, which we will discuss. The basis theorem that we use to prove these results also has some applications outside of Martin’s conjecture. We will use it to prove a few theorems related to Sacks’ question about whether it is provable in $\mathsf {ZFC}$ that every locally countable partial order of size continuum embeds into the Turing degrees. We will show that in a certain extension of $\mathsf {ZF}$ (which is incompatible with $\mathsf {ZFC}$ ), this holds for all partial orders of height two, but not for all partial orders of height three. Our proof also yields an analogous result for Borel partial orders and Borel embeddings in $\mathsf {ZF}$ , which shows that the Borel version of Sacks’ question has a negative answer. We will end the thesis with a list of open questions related to Martin’s conjecture, which we hope will stimulate further research. Abstract prepared by Patrick Lutz. E-mail: pglutz@berkeley.edu
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