Topics in Descriptive Set Theory

这个项目涉及集合论和描述集合论的研究，特别是涉及确定性公理的影响。 确定性公理是这样一种陈述，即每一个两人整数博弈都是确定的，也就是说，其中一个博弈者有一个获胜的策略。 尽管这一公理与选择公理相矛盾，选择公理是数学中一个被接受的部分，但它在20世纪60年代被提出来作为包含真实的数的集合论最小内部模型的合理假设。这个模型包含了在普通数学实践中出现的实数集合（例如，投射集及以后）。对这个模型的研究，反过来，给出了关于数学宇宙的直接信息。 标度的概念是描述集合论中的一个核心结构概念，确定性公理被用来发展投射集合的标度理论，后来扩展到整个模型。 然而，规模理论本身并不足以回答许多问题。 一个更详细的归纳分析是在20世纪80年代中期开始的，这是成功的，在较低层次的模型，包括投影集。 这个理论目前还没有扩展到整个模型，找到苏坎这样的扩展仍然是一个中心目标。 最近，一些新的研究方向已经展开，在某种程度上与这个模型的理论有关。 一个这样的方向涉及到阐明现有的理论之间的联系，这个模型和理论的超滤（希拉的p.c.f.理论）。 这些结果集中使用了伍丁的非定常理想理论，表明当人们从内部模型移动到真实的宇宙时，强坍缩原理是适用的。 究竟什么是被迫崩溃的，什么原则支配着这种现象，这是一个调查的主题。 发现和建立独立于完整归纳分析的原则也可能为通过整个模型传播分析的基本“骨架”提供一个框架。这个项目试图促进对集合的数学域的理解。 所有的数学都发生在这个宇宙中，这里的进步不仅在基础上很重要，而且因为它对数学的各个分支产生了直接的影响。 人们知道，回答许多基本的数学问题需要强有力的假设。 识别这些假设并探索它们的后果是集合论的一个主要主题。 确定性公理是这样一个公理的重要例子。这是在20世纪60年代制定的，但其后果的全部范围还不清楚。 最近的证据表明，它可能揭示了一些基本的数学问题，如连续体假设（有多少真实的数的问题）。

This project concerns research in set theory and descriptive settheory, particularly involving the influences of the axiom ofdeterminacy. The axiom of determinacy is the statement thatevery two-player integer game is determined, that is, one of theplayers has a winning strategy. Although this axiom contradictsthe axiom of choice, an accepted part of mathematics, it wasproposed in the 1960s to be a reasonable assumption for thesmallest inner model of set theory containing the real numbers.This model contains the sets of reals occurring in ordinarymathematical practice (e.g., the projective sets and beyond).The study of this model, in turn, gives direct information aboutthe mathematical universe. The notion of a scale is a centralstructural concept in descriptive set theory, and the axiom ofdeterminacy was used to develop the scale theory of theprojective sets, which was later extended throughout the entiremodel. The scale theory by itself, however, is not sufficient toanswer many questions. A more detailed inductive analysis wasbegun in the mid 1980s which was successful at the lower levelsof the model, including the projective sets. This theory doesnot currently extend through the entire model, and finding suchan extension remains a central goal. Recently, several new linesof investigation have opened up, relating in some way to thetheory of this model. One such direction concerns investigatingthe connections between the existing theory of this model and thetheory of ultrafilters (Shelah's p.c.f. theory). Preliminaryresults, which use centrally Woodin's theory of the nonstationaryideal, suggest strong collapsing principles apply as one movesfrom the inner model to the real universe. Exactly what isforced to collapse, and what principles govern this phenomenon,is a topic of investigation. Finding and establishing principlesindependent of the complete inductive analysis might also providea framework for propagating a basic "skeleton" of the analysisthrough the entire model.This project attempts to advance the understanding of themathematical universe of sets. All of mathematics takes placewithin this universe, and progress here is important not onlyfoundationally, but because of the direct influence on thevarious branches of mathematics. It has been known for some timethat strong assumptions are needed to answer many basicmathematical questions. Identifying these assumptions andexploring their consequences is a major theme in set theory. Theaxiom of determinacy is an important example of such an axiom.This was formulated in the 1960s but the full extent of itsconsequences is not known. Recent evidence suggests that it mayshed light on some basic mathematical questions, such as thecontinuum hypothesis (the question of how many real numbers thereare).