Set Theory

欧米伽逻辑是强制给出的自然逻辑：如果一个句子在V的所有强制扩张的所有列的初始段中都成立，则它是欧米伽有效的。如果存在一类真的Woodin基数，则给定的句子是否欧米茄有效本身是自一般不变的。欧米茄逻辑和欧米伽猜想的证明概念有一种自然的可能性，欧米伽猜想仅仅是这样一个猜想：如果有一类真的伍丁基数，那么对于任何句子，该句子是欧米伽可证明的当且仅当它是欧米伽有效的。欧米伽猜想的意义是，如果它是真的，那么人们可以完整地分析通过强迫所能达到的结果，而且欧米伽有效句子的集合可以用三阶数理论来定义。这限制了一般绝对的可能性。欧米茄猜想也是一般不变的，因此它不太可能是独立的，就像CH一样。因此，对欧米茄猜想的任何反驳都必须来自大的基数公理。欧米茄猜想适用于当前一代内部模型(扩展模型)。该方案的重点是通过检验超强以外大型基数的扩展器模型来考察一些大型基数假设反驳欧米茄猜想的可能性。这些扩展器模型不同于当前家族中允许在序列上使用长扩展器的情况。初步结果确定了几个关键点。首先，对于扩展器模型的标准推广到长扩展器的情况，一旦到达产生移动空间问题的扩展器序列，比较就失败(这个问题首先由Steel发现)。然而，存在一族扩展器模型，它有效地耗尽了所有已知的大基数公理，并且对于扩展器序列，不会出现移动空间问题(序列适当地短)。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。由此推论，没有任何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，那么假设V(可扩展基数)中适当的大基数，则欧米茄猜想一定在V中成立。最后，如果正如人们自然预期的那样，这些扩展模型有可定义的版本，那么就有许多真正深刻的推论。这些推论包括：如果存在可扩展基数，那么HOD正确地计算出适当类别的奇异基数的后继者。康托的连续统假设可以说是数学中最著名的不可解问题，这是希尔伯特从1900年以来的23个问题中的第一个问题清单。连续统假设简单地断言，任何无限的实数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，这个问题被证明是从集合论的公理中形式上无法解决的。科恩的这项开创性工作为集合论引入了一种新的技术--强迫方法。然而，连续体假设形式上不可解的事实并不一定意味着它不能解决。事实上，集合论过去40年的经验表明，一些形式上无法解决的问题是可以回答的。但在这些情况下使用的精确方法论不能解决连续统假设的问题。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。如果欧米茄猜想是真的，那么就有一个论点，或者至少是强有力的证据，认为连续统假设是假的。但与此无关的是，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证明欧米茄猜想是不可解的，因此我们有理由期待欧米伽猜想是否真的能被解决。对欧米茄猜想的任何反驳都必须来自大的基数公理。这项研究试图通过详细分析大型基数公理的层次来证明欧米茄猜想。初步的结果为这一方法的可行性提供了有力的证据。

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.