Randomizing Reals and the First-Order Consequences of Randoms
Randomizing Reals and the First-Order Consequences of Randoms
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随机化实数和随机数的一阶结果
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发表时间：
2014
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通讯作者：
Ian Haken
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作者：
Ian Haken
作者：Haken，Ian Robert|顾问：Slaman，西奥多A|翻译后摘要：在这篇论文中，我们研究了两个问题的算法随机性的主题。我们要解决的第一个问题是：“给定一个真实的，是否存在一个概率测度，对于这个概率测度，真实的不是一个原子，但相对于这个概率测度，真实的在算法上是随机的？这个问题最初是由Reimann和Slaman在Martin-Lof随机性方面进行研究的，本研究继续他们的调查，通过考虑更强的随机性概念的问题，并通过提供Reimann和Slaman的方法的元数学分析。我们调查的第二个问题是“2-随机实数存在的一阶后果是什么？“康纳利和斯拉曼表明，结果介于I-1和B-2之间，但留下了进一步分类的问题。我们证明了2-随机实数的存在并不意味着B ≠ 2，因此结论严格地位于I ≠ 1和B ≠ 2之间。此外，通过利用该证明中的方法，我们能够构造一个类κ模型，其中B 2失败，从而回答了Kaye在1995年提出的一个公开问题。
Author(s): Haken, Ian Robert | Advisor(s): Slaman, Theodore A | Abstract: In this dissertation we investigate two questions in the subject of algorithmic randomness. The first question we address is "Given a real, is there a probability measure for which the real is not an atom, but relative to which the real is algorithmically random?" This question was originally studied by Reimann and Slaman with respect to Martin-Lof randomness, and this research continues their investigation by considering the question with respect to stronger notions of randomness and by providing metamathematical analysis of Reimann and Slaman's methods.The second question we investigate is "What are the first-order consequences of the existence of 2-random reals?" Conidis and Slaman showed that the consequences lie somewhere between IΣ1 and BΣ2, but left open the question of further classification. We show that the existence of 2-random reals does not imply BΣ2, and thus the consequences lie strictly between IΣ1 and BΣ2. Furthermore, by utilizing the methods in this proof we are able to construct a κ-like model in which BΣ2 fails and thereby answer an open question posed by Kaye in 1995.