Set Theory

欧米伽逻辑是由强制给出的自然逻辑：如果一个句子在V的所有强制扩展的所有等级初始段中都成立，则该句子是欧米伽有效的。如果存在伍丁基数的真类，那么给定的句子是否欧米伽有效本身是一般不变的。欧米伽逻辑和欧米伽猜想的证明概念有一个自然的候选者，就是这样的猜想：如果存在一类沃丁基数，那么对于任何句子，当且仅当它是欧米伽有效的时，该句子才是欧米伽可证明的。欧米伽猜想的意义在于，如果它是真的，那么我们就可以完整地分析通过强制可以实现什么，而且欧米伽有效句子的集合是可以用三阶数论定义。这限制了泛型绝对性的可能性。欧米伽猜想也是泛型不变的，因此它不太可能像 CH 那样独立。因此，任何对欧米茄猜想的反驳都必须来自大的基本公理。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之外的大基数的扩展器模型来研究某些大基数假设反驳欧米伽猜想的可能性。这些扩展器模型与当前系列的不同之处在于序列上允许使用长扩展器。初步结果已经确定了几个关键点。首先，对于扩展器模型对长扩展器情况的标准推广，一旦到达出现移动空间问题的扩展器序列，比较就会失败（这个问题是第一个）然而，存在一系列扩展器模型，它们有效地穷举了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，我们会发现没有任何已知的大基数假说能够反驳欧米伽猜想。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 V 中存在适当的大基数（可扩展基数），则欧米伽猜想必须在 V 中成立。最后，如果正如人们自然预期的那样，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。这些推论包括以下内容：如果存在可扩展基数，则 HOD 正确计算奇异基数的正确类别的后继。连续统假说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。连续统假说简单地断言任何无限实数集合要么与整数等数，要么与所有实数的集合等数。大约 40 年前，这个问题被证明从集合公理中形式上无法解决理论。科恩的这部开创性著作向集合论引入了一种新技术，即强制方法。然而，连续统假设在形式上无法解决这一事实并不一定意味着它无法被解决。事实上，过去40年集合论的经验已经证明，一些形式上无法解决的问题是可以得到解答的。但在这些情况下使用的精确方法论无法解决连续统假说的问题。欧米伽猜想（大约10年前提出）自然产生于对强制方法的抽象分析，但在所谓的大基数公理的背景下。如果欧米茄猜想是正确的，那么就有一个论点，或者至少有强有力的证据，证明连续统假说是错误的。但与此无关，欧米伽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，因此可以合理地期望欧米茄猜想是否成立可以得到解决。对欧米茄猜想的任何反驳都必须来自大的基本公理。该研究项目旨在通过详细分析大基数公理的层次结构来证明欧米茄猜想。已经获得的初步结果为这种方法的合理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.