RUI: Incompleteness of the Third Kind in Set Theory

RUI：集合论中的第三类不完备性

• 批准号: 0501114
• 负责人: Maurice Stanley
• 金额: --
• 依托单位: San Jose State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501114&HistoricalAwards=false
• 关键词: RUI; Incompleteness; Third; Kind; Set

The PI proposes to continue work on several problems in set theory, using themethods of set and class forcing, infinite combinatorics, infinitary logic,and fine structure. Previous work has shown that certain combinatorialcharacterization problems do not have first-order solutions. For example, ingeneral there is no first-order definition of the set of subsets of omega-2that, in some omega-1 and omega-2 preserving outer model, have a closedunbounded subset. Other examples regarding subsets of other cardinals,branches through trees of certain sorts, and large homogeneous subsets forcertain partitions are known. One line of work concerns settling some furthercases. Another line of work concerns a more fundamental question. Can thisphenomenon can be mitigated by working relative to some reasonable extensionof ZFC, for example, one that is consistent with all large cardinal axioms.The logically most simple case of incompleteness of the third kind lies(necessarily) just beyond the scope of Woodin's celebrated genericabsoluteness theorem: Assume CH. Consider Sigma-2-1 sentences of analysiswith sets of reals as parameters. In general the set of such sentences thatare satisfiable in some outer model having the same reals is not first-orderdefinable. Does there exist an extension of ZFC that is consistent with alllarge cardinal axioms and relative to which this set is (lightface) Delta-2-2definable? Finally, the PI is interested in several questions regarding classforcing.The proposed work centers on incompleteness of a "third kind" in set theory.Incompleteness in set theory is important because all mathematics can beformalized in set theory. Propositions that are neither provable norrefutable from the axioms of set theory cannot be settledmathematically, at least in our current understanding. Goedel's famousIncompleteness Theorems show that such propositions exist. Sentencesdemonstrating this first kind of incompleteness formalize metamathematicalstatements. For example, the formalization of "ZFC is consistent" is neitherprovable nor refutable from the axioms of set theory (ZFC), provided thoseaxioms are, in fact, consistent. Even though "ZFC is consistent" is notprovable from ZFC, there is an obvious reason to favor it over itsnegation---studying mathematics within ZFC presupposes that ZFC is consistent.Incompleteness results proved using Cohen's method of (set) forcing representa second kind of incompleteness. Typically, given a "standard" model of ZFC,one constructs an outer model in which a given statement is true and one inwhich it is false. In the case of thissecond kind of incompleteness, there is often no reason to favor a statementor its negation. Deep work by Woodin, Steel, Martin, Foreman,and a number of others has suggested reasons to favor certain statements upthrough a certain level of logical complexity. Just beyond this level oflogical complexity lies incompleteness of a third kind. Here it is notpossible even to say which statements are satisfiable in some outer model."Characterization problems" are the combinatorial form of this phenomenon.Past work of the PI has highlighted that, in general, it is notpossible to characterize in set theory the "satiable objects" of certainsorts. Precise statements are technical, but an analogy gives the generalidea. In this analogy, the "objects" correspond to equations. An object is"sated" if the corresponding equation is solvable. An object is "satiable" ifthe corresponding equation is potentially solvable, that is, either solvableor solvable in some larger number system. In elementary mathematics, there isno reason to distinguish solvable and potentially solvable equations becausetypically there exist maximal number systems in which every potentiallysolvable equation of a particular sort is actually solvable. Such "maximalstandard models" do not exist in set theory. The analog of ananticharacterization result in set theory would be a type of equation forwhich there cannot be a good criterion for potential solvability.Anticharacterization is troubling for two reasons. First, in mathematics oneexpects that anything that is true is true for a good reason---so there oughtto be a criterion for insatiability. Secondly, in their strongest form, theseanticharacterization results hold only if the universe fails to be"sufficiently non-minimal". To some extent this threatens the well establishedthesis that all of mathematics is formalizable in first-order set theory,because non-minimality cannot be expressed in this language. The PIseeks to explore three aspects of anticharacterization. First, he seeks todetermine whether some specific cases of characterization problems aresolvable. Secondly, he seeks to discover whether adding auxiliary axioms tothe usual axioms of set theory might allow satiable objects to becharacterized. This could be construed as evidence in favor of these axioms.Finally, he seeks to continue work on abstract class forcing with an eyetowards understanding general outer models, at least in the presence ofconditions that render models highly non-minimal.

PI建议继续研究集合论中的几个问题，使用集合和类强迫，无限组合学，无限逻辑和精细结构的方法。 以前的工作已经表明，某些combinatorialcharacterization问题没有一阶的解决方案。 例如，一般没有一阶定义的集合的子集的欧米茄-2，在一些欧米茄-1和欧米茄-2保持外部模型，有一个封闭的无界子集。 关于其他基数的子集，通过某些种类的树的分支，以及某些分区的大型同质子集的其他例子是已知的。 其中一项工作涉及解决一些进一步的案件。 另一项工作涉及一个更根本的问题。这一现象是否可以通过相对于ZFC的某些合理的扩展来减轻，例如，与所有大的基数公理相一致的扩展。第三类不完备性的逻辑上最简单的情况（必然）恰好超出了Woodin著名的一般绝对性定理的范围：假设CH。考虑以实数集为参数的分析的Sigma-2-1句子。 一般来说，在具有相同实数的外部模型中可满足的句子的集合不是一阶可定义的。是否存在ZFC的一个扩展，它与alllarge基数公理相一致，并且相对于它，这个集合是（光面）Delta-2- 2可定义的？ 最后，PI对几个关于类强制的问题感兴趣。建议的工作集中在集合论中的“第三类”不完备性上。集合论中的不完备性很重要，因为所有的数学都可以在集合论中形式化。 从集合论的公理中既不能证明也不能反驳的命题不能用数学来解决，至少在我们目前的理解中是这样。哥德尔著名的不完全性定理证明了这种命题的存在。证明第一种不完全性的句子形式化了元数学陈述。例如，“ZFC是一致的”的形式化既不能从集合论（ZFC）的公理中证明，也不能从集合论（ZFC）的公理中反驳，只要这些公理实际上是一致的。尽管“ZFC是相容的”不能从ZFC中证明出来，但有一个明显的理由支持它而不是它的否定-在ZFC中研究数学的前提是ZFC是相容的。用Cohen的（集合）强迫方法证明的不完备性结果代表第二类不完备性。通常，给定ZFC的“标准”模型，人们构建一个外部模型，其中给定的语句为真，而另一个为假。 在第二种不完全性的情况下，通常没有理由赞成一个陈述或否定它。 伍丁、斯蒂尔、马丁、福尔曼和其他一些人的深入研究已经提出了在一定的逻辑复杂性水平上支持某些陈述的理由。在这种逻辑复杂性之上，存在着第三种不完整性。 在这里，甚至不可能说出哪些陈述在某个外部模型中是可满足的。“特征化问题”是这一现象的组合形式。PI过去的工作已经强调，一般来说，在集合论中不可能描述某些类型的“可满足对象”。 精确的陈述是技术性的，但类比给出了大致的概念。 在这个类比中，“对象”对应于方程。如果一个对象的相应方程是可解的，则该对象是“满足的”。一个对象是“可满足的”，如果相应的方程是潜在可解的，也就是说，或者可解的，或者在某个更大的数字系统中可解的。 在初等数学中，没有理由区分可解方程和潜在可解方程，因为通常存在极大数系统，其中每个特定类型的潜在可解方程实际上都是可解的。 这样的“最大标准模型”在集合论中并不存在。集合论中反特征化结果的类似物是一类方程，对于它的潜在可解性不存在一个好的判据。首先，在数学中，人们期望任何为真的东西都有一个好的理由-所以应该有一个永不满足的标准。 其次，在最强的形式下，这些反刻画结果只有在宇宙不是“充分非极小”的情况下才成立。在某种程度上，这威胁了所有数学都可以在一阶集合论中形式化的既定论点，因为非极小性不能用这种语言表达。 本书试图从三个方面探讨反特征化。首先，他试图确定是否一些具体的情况下，定性问题是可解决的。 其次，他试图发现是否添加辅助公理到集合论的通常公理可能允许satiable对象becharacterized。这可以被解释为支持这些公理的证据。最后，他试图继续研究抽象类强制，并着眼于理解一般的外部模型，至少在存在使模型高度非最小化的条件下。