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说法，即每个两人整数博弈都是确定的，也就是说，其中一个玩家有一个获胜的策略。尽管这一公理与数学中公认的选择公理相矛盾，但它在20世纪60年代被认为是包含实数的集合论最小内部模型的合理假设。这个模型包含了在正常数学实践中出现的实数集(例如，射影集和更多的实数集)。对这个模型的研究反过来又提供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中心结构概念，确定性公理被用来发展射影集的标度理论，该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。更详细的归纳分析是在1980年代中期开始的，它在模型的较低层次，包括射影集上取得了成功。这一理论目前还没有扩展到整个模型，找到这样的扩展仍然是一个中心目标。最近，有几条新的研究路线被开辟出来，在某种程度上与这一模型的理论有关。一个这样的方向是研究这个模型的现有理论和超滤器理论之间的联系(谢拉的P.C.F.理论)。主要使用伍丁的非定态理想理论的初步结果表明，当一个人从内部模型移动到真实宇宙时，强大的坍塌原理适用。到底是什么迫使崩溃，以及什么原则支配着这种现象，这是一个值得研究的话题。寻找和建立独立于完全归纳分析的原理也可能为在整个模型中传播分析的基本“骨架”提供一个框架。这个项目试图推进对集合的数学宇宙的理解。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，而且因为它直接影响到数学的各个分支。众所周知，许多基本的数学问题都需要强有力的假设来回答。识别这些假设并解释它们的结果是集合论的一个主要主题。确定性公理是这一公理的一个重要例子。这一公理是在20世纪60年代提出的，但其后果的全部程度尚不清楚。最近的证据表明，它可能揭示了一些基本的数学问题，例如连续统假设(有多少实数的问题)。

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).