Modal Model Theory

模态模型论将模态逻辑的思想注入到传统的数理逻辑模型论学科中。在这种观点下，人们将某一阶理论T的所有模型Mod(T)的类视为可能世界的Kripke模型。这里的世界是Mod(T)中的结构，在模型M中，陈述s是可能的，以防它扩展到另一个模型，其中s为真。同样，如果陈述s的所有扩展满足s，则陈述s在模型M上是必要的。本研究包括[1]中提出的一项努力，其中我们介绍了模态模型理论的主题和相关的基本定理。我们通过一个例子来证明这个主题的重要性——模态图理论——一个特别有见地的例子，说明了模态词汇的非凡力量。图论的模态语言可以表达连通性，k-可着色性，有限性，可数性，大小连续体，大小α - 1， α - 2, α -, α -, α -, α -, α -， α -不动点，α -超不动点等等。当且仅当一个图满足可数随机图理论时，它的参数服从极大性原则——每一个可能的必要陈述都已经为真；当且仅当它对有限图是全称时，它满足句子的极大性原则。为了清楚起见，我想把模态模型理论中出现的不同但密切相关的语言分开。1)我们用L表示理论t的语言2)L'是L在可能性和必然性以及布尔连接词（但不是量词）的模态操作符下的闭包。3) L”是全一阶模态语言，在模态运算符、布尔连接词和量词下闭合L。4) L''@用现实操作符@扩展了全模态语言，它允许人们引用现实世界。虽然很难预测这个项目的发展方向，但我的目的是进一步分析这些语言背景下出现的模态理论。例如，在[1]中，我们注意到中间模态语言L‘的许多属性不能扩展到全模态语言L’或L' @——这导致了我们想要研究的多个问题。特别是问题1。在ZFC中，是否可以定义所有图类的模态图论的满足关系？问题2。现实性在模态图理论中可表达吗？问题3。模态场理论中超越度是否可表示？问题4。我们是否能够将Ehrenfeucht-Fraïssé游戏推广到完整的模态语言中？然而，我的研究并不局限于b[1]中出现的问题和唯一的解决问题的方法。我认为这个项目不仅是对数理逻辑领域的贡献，而且，例如，作为加入围绕数学基础哲学的多元化辩论的讨论的尝试。事实上，模型理论的模态视角植根于[2]和[3]等作品，它们例证了哲学观点，即没有一个真正的终极数学宇宙；相反，有很多——所有这些都构成了数学上的多元宇宙。该项目属于EPSRC逻辑学和组合学研究领域。

Modal model theory injects ideas from modal logic into the traditional subject of model theory in mathematical logic. On this view, one treats the class of all models Mod(T) of some first-order theory T as a Kripke model of possible worlds. Worlds, here, are structures in Mod(T), and a statement s is possible at model M just in case it extends to another model whereat s is true. Similarly, statement s is necessary at model M if all its extensions satisfy s. This research includes an effort presented in [1], where we introduce the subject of modal model theory and relevant fundamental theorems. We demonstrate the importance of this subject by means of an example -- modal graph theory -- a particularly insightful case illustrating the remarkable power of the modal vocabulary. The modal language of graph theory can express connectedness, k-colourability, finiteness, countability, size continuum, size aleph-one, aleph-two, aleph-omega, beth-omega, first beth-fixed point, first beth-hyper-fixed-point, and much more. A graph obeys the maximality principle -- every possibly necessary statement is already true -- with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs.For clarity I should like to separate distinct but closely related languages arising in modal model theory.1) We denote by L the language of the theory T.2) L' is the closure of L under the modal operators of possibility and necessity as well as Boolean connectives (but not quantifiers).3) L'' is the full first-order modal language, closing L under modal operators, Boolean connectives and quantifiers.4) L''@ extends the full modal language with the actuality operator @, which allows one to refer to the actual world.While it is hard to predict the direction in which this project might develop, my intention is to further analyse modal theories arising in the context of these languages. For example, in [1], we note that many properties of the intermediate modal language L' do not extend to the full modal languages L'' or L''@ -- this led us to multiple questions that I should like to look at. In particular:Question 1. In ZFC, can one define the satisfaction relation for modal graph theory for the class of all graphs?Question 2. Is actuality @ expressible in modal graph theory?Question 3. Is transcendence degree expressible in modal field theory?Question 4. Can one generalise Ehrenfeucht-Fraïssé games to the full modal language, with actuality, L''@?My research, however, shall not be limited to issues arising in [1] and sole problem-solving. I view this project not only as a contribution to the field of mathematical logic but also, for example, as an attempt to join the discussion surrounding the debate on pluralism in the philosophy of mathematical foundations. Indeed, modal perspective on model theory has its roots in works such as [2] and [3], which exemplify the philosophical perspective that there is no one true and ultimate mathematical universe; rather there are many -- all comprising the mathematical multiverse as a whole.This project falls within the EPSRC Logic and combinatorics research area.