Mathematical Sciences: Determinacy, Inner Models, and Generic Embeddings

17发自：cwood@nsf.gov(Carol Wood)，注5/24/96 8：17am(2834字节：52 ln)至：jWhite ehu at nsf11，ahorton at nsf11 cc：cwood@nsf.gov，注主题：DuBose/Burke摘要(UNLV，9626212)下面是两个摘要(最后)。我将等待任何评论和建议等。在我的部分，我有点笼统，而不是特别指出，我对有这么多可测量的超过固定数量的Woodin基数的老鼠的确定性强度感兴趣。我想这很好，很笼统，更符合抽象的目的？？提前感谢您的任何意见，德里克------------------------------------------------------------------------DMS-9626212德里克·杜博斯/道格拉斯·伯克道格拉斯·伯克和德里克·杜博斯都参与了这个项目。道格拉斯·伯克的研究主要涉及集合论中的类嵌入问题。这些嵌入在使用新公理(大基数)来证明在集合论的通常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着广泛的应用。Burke在他的研究计划中有两个长期目标：(I)将大基数和泛型嵌入应用于组合问题(并将组合结果应用于关于泛型嵌入的问题)；(Ii)将大基数与实数的可定义良序之间的联系扩展到更大类别的可定义集合。德里克·杜波斯一直致力于建立在锐函数下封闭的内部模型与S类在分析体系底部附近的确定性之间的对应关系。他打算建立类似的对应关系，涉及分析层次中较高级别的类别。他还将继续研究适度的确定性假设以及小型“老鼠”和温和的大型基数性质的确定性强度。在过去的五十年里，许多自然的数学问题已经被证明是独立于集合论的通常公理的。为了确定这样的问题，引入了新的公理。特别重要的是大型基数公理和确定性假设。大的基数公理是关于无限的公理，而确定性假设表明某些可定义的无限博弈有获胜的策略。令人惊讶的是，这两组公理之间存在着强烈的联系。这两种方法都很重要，因为它们决定了许多关于实数集和其他“小”集的性质。

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.