Maximal Methods for Small Sets

小集的极大方法

• 批准号: 0401603
• 负责人: Paul Larson
• 金额: --
• 依托单位: Miami University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401603&HistoricalAwards=false
• 关键词: Maximal; Methods; Small; Sets

We intend to study maximal models for the powersets of the firsttwo uncountable cardinals as realized by the forcing method in thecontext of large cardinals and determinacy, and the applicationsof these models to other areas, especially topology. Many of theseissues complement a new theory of the infinite developed by W. Hugh Woodin called Omega-logic. By now it is a well establishedtheme in set theory that large cardinals impose certain forms ofcanonicity and absoluteness on the universe of sets. Inparticular, the existence of certain large cardinals implies thatthe theories of certain definable inner models of the universe areinvariant under forcing. Furthermore, these large cardinals alsotend to give rise to a detailed structure theory for these innermodels. The prototypical results of this type are results ofWoodin, building on work of Foreman, Magidor, Shelah, Martin andSteel, showing that a proper class of Woodin cardinals impliesthat the theory of the least inner model of set theory containingthe reals and the ordinals (L(R)) cannot be changed by setforcing, and that this fixed theory includes the Axiom ofDeterminacy. One natural program in the wake of these results isto identify and study larger models for which similar resultshold. Another direction, noting that the Axiom of Determinacycontradicts the Axiom of Choice, is to find similar forms ofabsoluteness compatible with AC. One way of doing this is toconsider statements to the effect that the universe of sets isclosed under certain forcing operations. Such statements aretypically called forcing axioms. Another approach is to considerforcing extensions of these inner models of determinacy. One majoradvance in this direction is Woodin's forcing Pmax. Heuristically,every natural question about the subsets of the first uncountablecardinal should have an answer in the Pmax extension of L(R).Nonetheless, there are several important questions about the Pmaxextension which remain open. Some of these questions concern theproperties of the nonstationary ideal on the first uncountablecardinal. One goal in pursuing these questions is to develop afiner analysis of the Pmax extension. In the other direction thereis the issue of whether results obtained by Pmax can be obtainedby other methods. Furthermore, the Pmax method has a number ofvariations, some of which have found application in topology.Cohen's method of forcing is a way of taking models of themathematical universe and producing larger, often very differentmodels. We intend to study properties of the first two uncountablecardinals as realized by the forcing method in the context of theregularity imposed by assuming the existence of large infiniteobjects (large cardinals) and certain regularity properties forset of real numbers (determinacy), and the applications of thesemodels to other areas, especially topology. Many of these issuescomplement a new theory of the infinite developed by W. HughWoodin. By now it is a well established theme in set theory thatlarge cardinals impose certain forms of canonicity andabsoluteness on the universe of sets. In particular, the existenceof certain large cardinals implies that the theories of certaindefinable inner models of the universe are invariant underforcing. Furthermore, these large cardinals also tend to give riseto a detailed structure theory for these inner models. One naturalprogram in the wake of these results is to identify and study larger models for which similar results hold. One way of doingthis is to consider statements to the effect that the universe ofsets is closed under certain forcing operations. Another approachis to consider forcing extensions of canonical inner models ofdeterminacy. One major advance in this direction is Woodin'sforcing Pmax. Heuristically, every natural question about thesubsets of the first uncountable cardinal should have an answer inthe Pmax extension. Nonetheless, there are several importantquestions about the Pmax extension which remain open. One goal inpursuing these questions is to develop a finer analysis of thePmax extension. In the other direction there is the issue ofwhether results obtained by Pmax can be obtained by other methods.Furthermore, the Pmax method has a number of variations, some ofwhich have found application in other areas of mathematics.

我们打算研究在大基数和确定性的背景下通过强迫方法实现的前两个不可数基数的幂集的极大模型，以及这些模型在其他领域，特别是拓扑学中的应用。许多这些问题补充了一个新的理论的无限发展的W。休·伍丁称之为欧米茄逻辑。到目前为止，在集合论中，大基数将某些形式的规范性和绝对性强加于集合论域，这是一个很好的主题。特别是，某些大基数的存在意味着某些可定义的宇宙内部模型的理论在强迫下是不变的。此外，这些大基数也往往会产生一个详细的结构理论，这些innermodels。 这种类型的原型结果是结果的Woodin，工作的基础上福尔曼，Magidor，Shelah，马丁和钢，表明一个适当的类的Woodin cardinals暗示，理论的最小内部模型的集合论containingthe reals和ordinals（L（R））不能改变由setforcing，这一固定的理论包括公理的决定性。在这些结果之后，一个自然的程序是识别和研究类似结果的更大模型。另一个方向，注意到确定性公理与选择公理相矛盾，是找到与AC相容的类似形式的绝对性。这样做的一种方法是考虑这样的陈述，即集合的宇宙在某些强制操作下是封闭的。这样的陈述通常被称为强制公理。另一种方法是对这些内在的确定性模型进行强制性扩展。在这个方向上的一个主要进展是Woodin的强迫Pmax。启发式地，每个关于第一不可数基数的子集的自然问题都应该在L（R）的Pmax扩张中有答案。其中一些问题涉及第一不可数切线上的非定常理想的性质。追求这些问题的一个目标是对Pmax扩展进行更深入的分析。在另一个方向上，有一个问题，即由Pmax得到的结果是否可以通过其他方法得到。此外，Pmax方法有许多变体，其中一些已经在拓扑学中得到了应用。科恩的强迫方法是一种采用数学宇宙模型并产生更大的、通常非常不同的模型的方法。我们打算研究前两个不可数基数的性质，通过强迫方法，在假设存在大的无限对象（大基数）和真实的数集的某些正则性（确定性）的情况下，以及这些模型在其他领域，特别是拓扑学中的应用。许多这些问题补充了一个新的理论的无限发展的W。HughWoodin到目前为止，在集合论中，大基数将某些形式的规范性和绝对性强加于集合论域，这是一个很好的主题。特别是，某些大基数的存在意味着某些可定义的宇宙内部模型的理论是不变的欠迫。此外，这些大的基数也倾向于为这些内部模型提供详细的结构理论。在这些结果之后，一个自然的程序是识别和研究更大的模型，其中类似的结果成立。这样做的一种方法是考虑声明的效果，宇宙的集合是封闭的某些强制操作。另一种方法是考虑强制扩展的典型内部模型的确定性。在这个方向上的一个主要进展是Woodin的强迫Pmax。启发式地，每个关于第一不可数基数子集的自然问题都应该在Pmax扩展中有答案。尽管如此，关于Pmax的扩展仍有几个重要的问题有待解决。追求这些问题的一个目标是对Pmax扩展进行更精细的分析。在另一个方向上，存在着用Pmax方法得到的结果是否可以用其他方法得到的问题。此外，Pmax方法有许多变体，其中一些已经在其他数学领域得到应用。