Mathematical Sciences: Determinacy, Inner Models, and Generic Embeddings

17来自：cwood@nsf.gov（Carol Wood）1996年5月24日上午8：17（2834字节：52 ln）收件人：jwhitehu，nsf 11，ahorton，nsf 11，抄送：cwood@nsf.gov，备注主题：DuBose/Burke摘要（UNLV，9626212）-消息内容- 文本项1：文本项 日期：1996年5月22日，星期三，14：58：28 -0700（PDT）发件人：DERRICK DUBOSE dubose@nevada.edu收件人：Carol Wood cwood@nsf.gov主题：摘要，油印版：1.0 亲爱的卡罗尔， 下面是两个摘要（最后）。我将等待任何意见，建议，等对我的一部分，我有点一般，而不是特别指出，特别是，我感兴趣的确定性强度的小鼠有这么多的可测量以上的一些固定数量的伍丁红雀。我想这很好，是一般的，更好地满足了抽象的目的？？ 提前感谢您的任何评论， 井架 ------------------------------------------------------------------------ DMS-9626212 Derrick DuBose/道格拉斯伯克 道格拉斯伯克和德里克杜博斯参与了这个项目。 道格拉斯伯克的研究主要涉及集合论中的类属嵌入。 这些嵌入在使用新的公理（大基数）来证明集合论中通常的基本公理中不可判定的陈述方面起到了重要作用。 特别是，它们在描述集合论和组合集合论中有许多应用。伯克有两个长期的目标，在他的研究计划：（一）适用于大基数和通用嵌入组合问题（并适用于组合结果的问题，通用嵌入），以及（ii）扩大之间的联系大基数和可定义良序的真实的号码更大类的可定义集。Derrick DuBose一直致力于建立在尖锐函数下封闭的内部模型与分析层次底部附近类的确定性之间的对应关系。 他打算建立类似的对应关系，涉及层次分析中更高的类。 他还将继续研究适度的确定性假设和确定性强度的小“老鼠”和温和的大基数属性。 在过去的五十年里，许多自然数学问题已经被证明是独立于集合论的通常公理。 为了解决这些问题，引入了新的公理。 特别重要的是大基数公理和决定性假设。大基数公理是无穷公理，而确定性假说则认为某些可定义的无穷博弈具有获胜策略。 令人惊讶的是，这两组公理之间存在着很强的联系。两者都很重要，因为它们决定了关于真实的数集合和其他"小“集合的许多性质。

17 From: cwood@nsf.gov (Carol Wood) at NOTE 5/24/96 8:17AM (2834 bytes: 52 ln) To: jwhitehu at nsf11, ahorton at nsf11 cc: cwood@nsf.gov at NOTE Subject: DuBose/Burke abstract (UNLV, 9626212) ------------------------------- Message Contents ------------------------------- Text item 1: Text Item Date: Wed, 22 May 1996 14:58:28 -0700 (PDT) From: DERRICK DUBOSE dubose@nevada.edu To: Carol Wood cwood@nsf.gov Subject: abstracts Mime-Version: 1.0 Dear Carol, below are the two abstracts (finally). I shall be waiting for any comments, suggestions, etc. On my portion, I was somewhat general, instead of specifically indicating that in particular, I am interested in the determinacy strength of mice with so many measurables above some fixed number of Woodin cardinals. I guess this is fine, being general, and better satisfies the purpose of the abstract?? Thanks in advance for any comments, Derrick ------------------------------------------------------------------------ DMS-9626212 Derrick DuBose/Douglas Burke Douglas Burke and Derrick DuBose are involved in this project. The research of Douglas Burke is primarily concerned with generic embeddings in set theory. These embeddings have been instrumental in using new axioms (large cardinals) to prove statements that are undecidable in the usual, basic axioms of set theory. In particular, they have many applications in descriptive set theory and combinatorial set theory. Burke has two long term goals in his research program: (i) to apply large cardinals and generic embeddings to combinatorial questions (and apply combinatorial results to questions about generic embeddings), and (ii) to extend the connection between large cardinals and definable well-orderings of the real numbers to larger classes of definable sets. Derrick DuBose has been engaged in establishing correspondences between inner models closed under sharp functions and the determinacy of classe s near the bottom of the analytic hierarchy. He intends to establish similar correspondences involving classes higher up in the analytic hierarchy. He will also continue to investigate moderate determinacy assumptions and the determinacy strength of small "mice" and of mild large cardinal properties. During the last fifty years, many natural mathematical questions have been shown to be independent of the usual axioms of set theory. In order to decide such questions, new axioms have been introduced. Of particular importance are Large Cardinal Axioms and Determinacy Hypotheses. The Large Cardinal Axioms are axioms of infinity, whereas determinacy hypotheses state that certain definable infinite games have winning strategies. Surprisingly, a strong connection exists between these two groups of axioms. Also both are important in that they decide many properties about sets of real numbers and other ``small'' sets.