RUI: Incompleteness of the Third Kind in Set Theory

PI建议继续研究集合论中的几个问题，使用集合类强迫、无限组合学、无限逻辑和精细结构的方法。前人的工作已经表明，某些组合刻画问题没有一阶解。例如，在某些保持omega-1和omega-2的外部模型中，一般不存在具有闭无界子集的omega-2的子集的集合的一阶定义。关于其他基数的子集、通过某些种类的树的分支、以及用于某些划分的大的同质子集的其他例子是已知的。其中一项工作是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。这种现象可以通过相对于ZFC的一些合理的扩展来缓解吗，例如，一个与所有大型基数公理一致的扩展。逻辑上最简单的第三类不完全性的情况(必然)正好超出了伍丁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Sigma-2-1分析句子。一般而言，在某些具有相同实数的外部模型中可满足的这类句子的集合不是一阶可定义的。是否存在与所有大型基数公理一致的ZFC扩展，并且相对于该扩展，该集合是(亮面)Delta-2-2可定义的？最后，PI对与分类强化有关的几个问题感兴趣。建议的工作集中在集合论中第三类的不完全性。集合论中的不完全性是重要的，因为所有的数学都可以在集合论中形式化。从集合论的公理来看，既不能证明也不能反驳的命题，至少在我们目前的理解中是不能用数学方法解决的。歌德尔著名的不完全性定理表明，这样的命题是存在的。证明第一种不完备性的意义使元数学语句形式化。例如，如果集合论公理(ZFC)实际上是一致的，那么“ZFC是一致的”的形式化既不能证明，也不能反驳。尽管ZFC不能证明ZFC是一致的，但有一个明显的理由支持它而不是它的否定-在ZFC内学习数学的前提是ZFC是一致的。用科恩的(集)强迫方法证明的不完全性结果代表了第二种不完备性。通常，在给定ZFC的“标准”模型的情况下，人们会构建一个外部模型，在该模型中，给定的语句为真，而其中的语句为假。在第二种不完备性的情况下，通常没有理由支持一个陈述或它的否定。伍丁、斯蒂尔、马丁、福尔曼和其他一些人的深入研究表明，有理由通过一定程度的逻辑复杂性来支持某些陈述。在这种逻辑复杂性的基础上，存在着第三种不完备性。在这里，甚至不可能说哪些陈述在某些外部模型中是可满足的。“特征问题”是这种现象的组合形式。PI过去的工作强调，一般而言，不可能在集合论中刻画某些种类的“可满足对象”。精确的陈述是技术性的，但类比给出了一个概念性的概念。在这个类比中，“对象”对应于方程式。如果相应的方程是可解的，则一个对象是“满足的”。如果相应的方程是潜在可解的，即在一些较大的数系统中是可解的或可解的，则一个对象是“可满足的”。在初等数学中，没有理由区分可解方程和潜在可解方程，因为在典型的极大数系统中，每一个特定类型的潜在可解方程实际上都是可解的。这种“最大标准模型”在集合论中是不存在的。类似于集合论中的反刻画结果将是一类方程，其潜在可解性不存在良好的判据。反刻画令人担忧有两个原因。首先，在数学中，人们期望任何真实的东西都是真实的，这是有充分理由的-所以应该有一个衡量贪得无厌的标准。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。这在某种程度上威胁到了所有数学都可以在一阶集合论中形式化这一根深蒂固的论点，因为非极小性不能用这种语言来表达。PI试图探索反特色化的三个方面。首先，他试图确定一些具体的刻画问题的案例是否可以解决。其次，他试图发现，在集合论的通常公理中加入辅助公理，是否可以使令人满足的对象被刻画。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迫，着眼于理解一般的外部模型，至少在存在使模型高度非极小的条件的情况下。

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.