论文标题
模态模型理论
Modal model theory
论文作者
论文摘要
我们介绍了模态模型理论的主题，其中人们在扩展概念下研究了一类类似结构的数学结构，从而引起了数学上自然的可能性和必要性概念。如果$φ$在该结构的某个扩展中为真，则可以在结构中（书面$ \diamondφ$）中使用语句$φ$，如果在结构的所有扩展中都是正确的，则需要$φ$（书面$ \boxφ$）。对我们来说，一个主要案例将是给定理论t的所有模型的类（t） - 所有图，所有图，所有组，所有字段或您所考虑的 - 在子结构关系中所考虑的。在本文中，我们旨在开发产生的模态模型理论。 The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, $k$-colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_ω$, $\beth_ω$, first $\beth$-fixed point, first $ \ beth $ -yper固定点等等。一个图遵守最大原理$ \ diamond \boxφ（a）\toφ（a）$，当且仅当它满足可计数随机图的理论时，并且仅在有限图的通用时才能满足句子的最大原理。
We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement $φ$ is possible in a structure (written $\Diamondφ$) if $φ$ is true in some extension of that structure, and $φ$ is necessary (written $\Boxφ$) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) of all models of a given theory T---all graphs, all groups, all fields, or what have you---considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, $k$-colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_ω$, $\beth_ω$, first $\beth$-fixed point, first $\beth$-hyper-fixed-point and much more. A graph obeys the maximality principle $\Diamond\Boxφ(a)\toφ(a)$ 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.
