Set Theory

集合论

• 批准号: 0355334
• 负责人: William Woodin
• 金额: $ 38.03万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355334&HistoricalAwards=false
• 关键词: Set; Theory

Omega-logic is the natural logic given by forcing:a sentence is Omega-valid if it holds in all rankinitial segments of all forcing extensions of V. Ifthere exists a proper class of Woodin cardinals thenwhether or not a given sentence is Omega-valid is itselfgenerically invariant. There is a natural candidateof the notion of proof for Omega-logic and the Omega-Conjectureis simply the conjecture that if there is a proper classof Woodin cardinals then for any sentence, the sentenceis Omega-provable if and only if it is Omega-valid.The significance of the Omega-Conjecture isthat if it is true then one has a complete analysis ofwhat can be achieved by forcing and moreover the setof Omega-valid sentences is definable in third ordernumber theory. This places a limit on the possibilitiesfor generic absoluteness.The Omega-Conjecture is also generically invariant andso it is unlikely to be independent is the same fashionthat CH is. Therefore any refutation of the Omega-Conjecture must come from large cardinal axioms. The Omega-Conjecture holds in the current generation of inner models (extender models). The focus of the proposal is to investigate the possibilitythat some large cardinal hypothesis refutes theOmega-Conjecture by examining extender models for large cardinals beyond superstrong.These extender models differ from the current familyin that long extenders are allowed on the sequence.Preliminary results have established several key points.First, for the standard generalization of extender modelsto the case of long extenders, comparison fails as soon asone reaches sequences of extenders for which the movingspaces problem arises (this problem was first identified by Steel).Nevertheless there is family of extender models which effectively exhaust all known large cardinal axioms and for theseextender sequences the moving spaces problem does not arise (the sequences are suitably short). If comparisoncan be established for these models then Omega-Conjectureholds in these inner models. As a corollary one would obtainthat no known large cardinal hypothesis can refutethe Omega-Conjecture. Finally if a suitable iterationhypothesis holds in these inner models then assumingappropriate large cardinals in V (extendible cardinals)the Omega-Conjecture must hold in V. Finally if, as onemight naturally expect, there are definable versions of these extendermodels then there are a number of truly profound corollaries.These include the following: if there is an extendible cardinal then HOD correctly computes the successor for a proper class of singular cardinals.Cantor's Continuum Hypothesis is arguably the most famousunsolvable problem in Mathematics, it was the first problemon Hilbert's list of 23 problems from 1900. The Continuum Hypothesissimply asserts that any infinite collection of real numbersis either equinumerous with the integers or equinumerouswith the collection of all real numbers.Around 40 years ago this problem was shown to be formallyunsolvable from the axioms of Set Theory. This seminal workof Cohen introduced a new technique to Set Theory, the method of forcing. However the fact that the Continuum Hypothesisis formally unsolvable does not necessarily imply thatit cannot be solved. Indeed the experience in Set Theoryover the last 40 years has demonstrated that some questionswhich are formally unsolvable can be answered. But theprecise methodology used in these cases cannot workfor the problem of the Continuum Hypothesis.The Omega-Conjecture (proposed around 10 years ago) arisesnaturally from an abstract analysis of the method of forcingbut in the context of so called large cardinal axioms. Ifthe Omega-Conjecture is true then there is an argument, or atleast strong evidence, that the Continuum Hypothesis is false. But independent of this, whether or not the Omega-Conjecture is true has profound consequences for the foundations of Set Theory. Cohen's method of forcing cannot be used to establish thatthe Omega-Conjecture is unsolvable so it is reasonable toexpect that whether or not the Omega-Conjecture is truecan be resolved. Any refutation of the Omega-Conjecturemust come from large cardinal axioms. This research projectconcerns trying to prove the Omega-Conjecture by a detailedanalysis of the hierarchy of large cardinal axioms. Preliminaryresults have been obtained which offer strong evidence forplausibility of this approach.

Omega-逻辑是由强制给出的自然逻辑：一个句子是Omega-有效的，如果它在V的所有强制扩张的所有秩初始段中成立。如果存在一个适当的Woodin基数类，那么给定的句子是否是Omega-有效的是它自身一般不变的。有一个自然的候选人的概念，证明欧米茄逻辑和欧米茄猜想只是猜想，如果有一个适当的类的伍丁基数，然后对任何句子，这个猜想是Omega-可证明的当且仅当它是Omega-有效的。Omega-猜想的意义在于，如果它是真的，那么人们就有了一个完整的分析，可以通过强迫实现什么，而且Omega-有效句子在三阶数论中是可定义的。这就限制了泛属绝对性的可能性。欧米茄猜想也是泛属不变的，所以它不太可能像CH一样是独立的。因此，任何对欧米茄猜想的反驳都必须来自大基数公理。欧米茄猜想在当前一代的内部模型（扩展模型）中成立。该提案的重点是通过检查超强以上的大基数的扩展器模型来研究某些大基数假设反驳Omega猜想的可能性。这些扩展器模型与当前家族的不同之处在于允许序列上的长扩展器。初步结果建立了几个关键点。首先，对于扩展器模型到长扩展器的情况的标准推广，比较失败，因为一旦到达序列的扩展器，其中移动空间的问题出现（这个问题首先确定钢）。然而，有家庭的扩展器模型，有效地用尽所有已知的大基数公理和theseextender序列的移动空间的问题不会出现（序列是适当的短）。如果可以对这些模型进行比较，那么欧米茄猜想在这些内部模型中成立。作为一个推论，人们会得到，没有已知的大基数假设可以反驳欧米茄猜想。最后，如果一个合适的迭代假设在这些内部模型中成立，则在V中假设适当的大基数最后，如果，正如人们可能自然期望的那样，这些扩展模型有可定义的版本，那么就有一些真正深刻的推论。这些推论包括：如果有一个可扩展基数，那么HOD正确地计算出一个适当的奇异基数类的后继。康托的连续统假设可以说是数学中最著名的不可解问题，这是希尔伯特1900年列出的23个问题中的第一个。连续统假说简单地断言，任何真实的数的无限集合要么与整数相等，要么与所有真实的数的集合相等。科恩的这部开创性著作为集合论引入了一种新的技术，即强迫方法。然而，连续统假说形式上不可解的事实并不一定意味着它不能被解决.事实上，集合论在过去40年中的经验已经证明，一些形式上无法解决的问题是可以回答的。但是，在这些情况下使用的精确的方法不能工作的连续统假设的问题。欧米茄猜想（提出了大约10年前）从一个抽象的分析的方法，强迫，但在所谓的大基数公理的背景下自然。如果欧米茄猜想是真的，那么就有一个论点，或者至少是强有力的证据，证明连续统假设是假的。但与此无关，欧米茄猜想是否正确对集合论的基础有着深远的影响。科恩的强迫方法不能用来确定Ω猜想是不可解的，因此有理由期待Ω猜想是否正确可以得到解决。对欧米茄猜想的任何反驳都必须来自大的基本公理。本研究计划通过对大基数公理的层次结构的详细分析来证明Ω猜想。已获得的结果提供了强有力的证据证明这种方法的可行性。