Stability Theory in Continuous First Order Logic

一阶逻辑适用于描述和研究“离散”结构类：域、群、图等。模型理论，特别是稳定性理论，为这类提供了几个引人注目的结构定理：例如，每个向量空间都是由（线性基的）基数决定的，而每个代数闭域都是由超越基决定的，这两个都是莫利定理的特殊情况：在这个定理适用的任何一类结构中，所有结构都是由一个合适的基生成的。希拉的分类理论是对莫利定理的广泛推广，为更多种类的结构提供了结构定理。虽然所有这些理论都是针对离散结构发展起来的，但至少在某种程度上，它们似乎也适用于度量结构或“连续”结构：例如，将morley定理与每个希尔伯特空间（作为一个完整的度量向量空间）由一个标准正交基生成的事实进行比较。本文旨在进一步将稳定性和分类理论的经典结构结果应用于连续结构的分类。因此，本提案寻求将稳定性理论的结果和技术，特别是超稳定性理论，扩展到连续一阶逻辑的设置，旨在推广shelah的主间隙定理。连续一阶逻辑的优势在于它是一阶逻辑的自然推广，同时它还可以容纳在泛函分析和概率论中产生的许多自然类（度量）结构(可能带有附加结构的各种巴纳赫空间、概率空间和自适应空间的度量代数、由于非离散度量的存在而产生的新复杂性要求我们修改基本定义（超稳定性，秩等），并使大多数经典理论，特别是正则类型的概念，似乎不适用。尽管如此，最近在这个方向上已经取得了相当大的进展，例如将拉克伦定理应用于连续超稳定理论，并且有迹象表明，类似的技术可以用于研究权的有限性及其结果，这似乎是实现该计划的自然下一步。

First order logic is suitable for describing and studying classes of"discrete" structures: fields, groups, graphs, etc. Model theory, andin particular stability theory, provided several striking structuretheorems for such classes: for example, the fact that every vectorspace is determined by (the cardinality of) a linear base, while everyalgebraically closed fields is determined by a transcendence base, areboth special cases of Morley's Theorem: in any class of structure towhich this theorem applies, all structures are generated by a suitablebase. Shelah's classification theory is a vast generalisation ofMorley's Theorem, yielding structure theorems to many more classes ofstructures. While all these theories were developed for classes ofdiscrete structures, they seems to hold, at least to some extent, forclasses of metric, or "continuous" structures: for example, compareMorley's theorem with the fact that every Hilbert space is generated(as a complete metric vector space) by an orthonormal base. Thisproposal seeks to further adapt classical structure results fromstability and classification theory to classes of continuousstructures.This proposal therefore seeks to extend the results and techniques ofstability theory, and in particular superstability, to the setting ofcontinuous first order logic, aiming towards a generalisation ofShelah's Main Gap Theorem. Continuous first order logic has theadvantage of being a natural generalisation of first order logic,while at the time accommodating many natural classes of (metric)structures arising in functional analysis and probability theory(various classes of Banach spaces possibly with additional structure,measure algebras of probability spaces and of adapted spaces, etc.)New complications arising from the presence of a non-discrete metricrequire us to revise the fundamental definitions (superstability,ranks, etc.), and renders most of the classical theory, and inparticular the notion of regular types, seemingly inapplicable.Nevertheless there has been considerable progress recently in thisdirection, such as an adaptation of Lachlan's theorem to continuoussuperstable theories, and there are indications that similartechniques can be used towards finiteness of weight and itsconsequences, which seem the natural next step towards the achievementof this programme.