Maximal Methods for Small Sets

我们打算研究在大基数和确定性的情况下用强迫方法实现的前两个不可数基数的幂集合的极大模型，以及这些模型在其他领域，特别是拓扑学中的应用。这些问题中的许多都是对W.Hugh Woodin发展的一种新的无限理论的补充，该理论称为欧米茄逻辑。到目前为止，大基数将某些形式的正则性和绝对性强加于集合的宇宙，这是集合论中一个根深蒂固的主题。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，这些大的基数也倾向于产生这些内部模型的详细结构理论。这种类型的典型结果是Woodin在Foreman、Magidor、Shelah、Martin和Steel工作的基础上得出的结果，表明一类合适的Woodin基数意味着包含实数和序数的集合论的最小内模理论(L(R))不能通过集合强迫来改变，并且这个固定的理论包括决定公理。在这些结果之后，一个自然的计划是识别和研究更大的模型，这些模型应该有类似的结果。另一个方向，注意到决定性公理与选择公理相矛盾，是找到与AC兼容的类似形式的绝对。要做到这一点，一种方法是考虑这样的说法，即集合的宇宙在某些强迫操作下是封闭的。这种说法通常被称为强制公理。另一种方法是考虑强制扩展这些确定性的内部模型。在这个方向上的一个主要进步是伍丁的强迫Pmax。启发式地，在L(R)的Pmax扩张中，每个关于第一个不可数扩张的子集的自然问题都应该有一个答案。然而，关于Pmax扩张，有几个重要的问题仍然是开放的。其中一些问题涉及第一个不可数理想上的非定常理想的性质。追求这些问题的一个目标是开发对Pmax扩展的更精细的分析。在另一个方向上，存在由Pmax获得的结果是否可以通过其他方法获得的问题。此外，Pmax方法有许多变体，其中一些已经在拓扑学中得到应用。科恩的强迫方法是一种获取数学宇宙的模型并产生更大的、通常非常不同的模型的方法。我们打算研究在假设大无限对象(大基数)的存在和实数集(确定性)的某些正则性的情况下，用强迫方法实现的前两个不可数基数的性质，以及这些模型在其他领域，特别是拓扑学中的应用。这些问题中的许多都是对W.HughWoodin发展的无限的新理论的补充。到目前为止，大基数将某些形式的正则性和绝对性强加于集合的宇宙，这是集合论中一个公认的主题。具体地说，某些大基数的存在意味着可确定的宇宙内部模型理论是不变的。此外，这些大基数还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，一个自然的计划是识别和研究类似结果适用的更大的模型。要做到这一点，一种方法是考虑这样的陈述，即集合的宇宙在某些强迫操作下是封闭的。另一种考虑强制扩展确定性规范内部模型的方法。在这个方向上的一个重大进步是伍丁的强迫Pmax。启发式地，每一个关于第一个不可数基数的子集的自然问题都应该在Pmax扩张中有一个答案。尽管如此，关于Pmax扩展的几个重要问题仍然悬而未决。追求这些问题的一个目标是开发对Pmax扩展的更精细的分析。另一个方向是Pmax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了应用。

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.