Set Theory

欧米茄逻辑是由强迫给出的自然逻辑：一个句子是欧米茄有效的，如果它在所有v的强迫扩展的所有秩初始段中都成立。如果存在一个适当的伍丁基数类，那么一个给定的句子是否欧米茄有效本身是一般不变的。欧米茄逻辑的证明概念有一个自然的候选者，欧米茄猜想只是一个猜想，如果有一个适当的伍丁基数类别，那么对于任何句子，当且仅当它是欧米茄有效的，这个句子是欧米茄可证明的。欧米茄猜想的意义在于，如果它是真的，那么人们就有了一个完整的分析，可以通过强迫来实现什么，而且欧米茄有效句子的集合在三阶数理论中是可定义的。这就限制了泛型绝对性的可能性。ω -猜想也是一般不变的，所以它不可能像CH那样独立。因此，对欧米茄猜想的任何反驳都必须来自大的基本公理。欧米茄猜想持有在当前一代的内部模型（扩展模型）。该提案的重点是通过检查超大基数的扩展器模型来研究一些大型基数假设驳斥ω -猜想的可能性。这些扩展器模型不同于当前的家族，因为序列上允许长扩展器。初步结果确定了几个关键点。首先，对于扩展器模型的标准泛化到长扩展器的情况，一旦到达移动空间问题出现的扩展器序列（这个问题首先由Steel发现），比较就会失败。然而，有一组扩展器模型可以有效地耗尽所有已知的大基本公理，并且对于这些扩展器序列不会出现移动空间问题（序列适当短）。如果可以为这些模型建立比较，那么omega - conjecture就适用于这些内部模型。作为一个推论，人们会得到，没有已知的大的基本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，那么假设在V中适当的大基数（可扩展基数）欧米茄猜想必须在V中成立。最后，如果，正如人们可能自然期望的那样，这些扩展模型存在可定义的版本，那么就会有许多真正深刻的推论。这些包括：如果有一个可扩展基数，那么HOD正确地计算一个适当类的奇异基数的后继。康托的连续统假设可以说是数学中最著名的无法解决的问题，它是希尔伯特1900年列出的23个问题中的第一个问题。连续统假设简单地断言，任何实数的无限集合要么与整数相等，要么与所有实数的集合相等。大约40年前，这个问题被证明是无法从集合论公理中正式解决的。科恩的这项开创性工作为集合论引入了一种新技术，即强迫方法。然而，连续统假设在形式上不可解的事实并不一定意味着它不能解。事实上，集合论在过去40年的经验已经证明，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方法不适用于连续统假设的问题。欧米茄猜想（大约10年前提出）自然地产生于对强迫方法的抽象分析，但在所谓的大基本公理的背景下。如果欧米茄猜想是正确的，那么就有一个论点，或者至少是强有力的证据，证明连续统假说是错误的。但除此之外，omega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证明欧米茄猜想是不可解的，所以我们有理由期待欧米茄猜想是否正确可以被解决。对欧米茄猜想的任何反驳都必须来自大的基本公理。本研究项目旨在通过对大基本公理的层次结构的详细分析来证明欧米茄猜想。初步结果为该方法的可行性提供了强有力的证据。

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.