Potentialist Systems for Set Theory and Beyond

强迫是现代集合论中最重要的技术之一。它提供了一种高度通用的方法，可以从旧的集合理论模型中生成新的模型。通常，强迫被用作获得一致性和独立性结果的工具；然而，它本身也是数学兴趣的对象。任何特定的强迫概念都会在集合论的模型类上引出一个关系，如果一个模型是另一个模型的强迫扩展，则两个模型是相关的。Andrzej Mostowski证明了任何有限偏序集都可以嵌入到一个模型M的泛型多元宇宙中，即由这些模型组成的子结构，这些模型可以通过强迫-扩展关系从M中获得。利用“集合论区块链”的新技术，Joel Hamkins最近加强了这一结果，包括了广泛的无限序集，以及建立了许多其他的结构性质。实际上，M的泛型多重宇宙类似于图灵度的结构，它也具有嵌入后的性质。这项研究计划的目的之一是调查这种联系的程度。例如，泛型多元宇宙是否表现出嵌入-扩展特性：一个传票集的任何嵌入都可以扩展到整个偏序集的嵌入？强迫-可拓关系是潜在系统的一个例子：一些理论的一类模型以及与这些模型相关的可拓概念。拟议的研究方案也将调查这一更普遍的领域。进一步的可拓概念可以强加于集合论模型的类，如末端可拓的概念，和其他自然类别的模型也可以被考虑。后者的一个富有成果的例子是算术的所有模型类，其中许多扩展概念都是适用的，并且与集合论的情况密切相关。w·休·伍丁最近发现了算术的通用算法。该算法列举了一个有限序列，并且具有一个显著的性质，即如果在算术模型M中列举了s，并且t是扩展s的有限序列，则存在M的一个末端扩展，其中枚举了t。最近，Joel Hamkins在集合论的可数模型中发现了一个类似的结果：在右末端扩展中存在一个有限集合，由E2公式指定，它可以包含任何想要的集合。这两个定理揭示了公式和模型之间深刻的新相互作用，并为一系列应用打开了大门。在算术方面，一个结论是，在每一个算术模型中，都存在一个丢芬图方程，它没有解，但在某个末端扩展中得到解。在集合论方面，平行的结果是不存在具有极大e2图的集合论可数模型。这方面的研究方向很多。如果去掉模型可数的要求，结果能得到加强吗？是否存在其他类型的全称算法和全称有限集，例如，关于强制扩展的全称集？其他各种各样的理论提供了势系统的有趣例子，以及与各种数学领域联系的可能性。子图包含下的所有图类提供了一个自然的例子。这可以用模态逻辑来研究，模态逻辑是一种用关系来解释的逻辑。图论势系统的一阶模态逻辑是相当强大的；例如，可以表示图的双色性，甚至可以表示图的有限性和可数性。这一领域的主要目标是确定这种逻辑的精确强度。该项目属于EPSRC逻辑学和组合学研究领域。

Forcing is one of the most important techniques of modern set theory. It provides a highly versatile method of generating new models of set theory from old. Typically, forcing is used as a tool for obtaining consistency and independence results; however it is also an object of mathematical interest in its own right. Any particular notion of forcing induces a relation on the class of models of set theory, two models being related if one is a forcing extension of the other. Andrzej Mostowski proved that any finite poset can be embedded, in a strong sense, into the generic multiverse of a model M: the substructure consisting of those models accessible from M via the forcing-extension relation. Using the novel technique of the 'set-theoretic blockchain', Joel Hamkins has recently strengthened this result to include a wide class of infinite posets, as well as establishing many other structural properties. In fact, the generic multiverse of M resembles the structure of the Turing degrees, which also enjoys the poset-embedding property. One of the aims in this research proposal is to investigate how far this connection goes. For example, does the generic multiverse exhibit the embedding-extension property: that any embedding of a subposet can be extended to an embedding of the whole poset?The forcing-extension relation is one example of a potentialist system: a class of models of some theory together with an extension concept relating these models. The proposed research programme will also investigate this more general area. Further extension concepts may be imposed on the class of set theoretic models, such as the notion of end-extension, and other natural classes of models may also be considered. One fruitful example of the latter is the class of all models of arithmetic, to which numerous extension concepts are applicable, and which enjoys close connections with the set theory case. W. Hugh Woodin recently discovered the universal algorithm for arithmetic. This algorithm enumerates a finite sequence, and has the remarkable property that if it enumerates s in a model M of arithmetic, and t is a finite sequence extending s, then there is an end-extension of M in which it enumerates t. Recently, Joel Hamkins has found an analogous result for countable models of set theory: there is a finite set, specified by a E2 formula, which can be made to include any set desired, in the right end-extension. These two theorems uncover a deep new interplay between formulae and models, and open the door to a range of applications. On the arithmetical side, one consequence is that in every model of arithmetic, there is a Diophantine equation which has no solution, but which gains a solution in some end-extension. On the set-theoretic side, the parallel result is that there is no countable model of set theory having a maximal E2-diagram. There are many research directions in this area. Can the result be strengthened by removing the requirement that the models be countable? Do other kinds of universal algorithms and universal finite sets exist, for example, sets universal with respect to forcing-extensions?Various other theories provide interesting examples of potentialist systems, along with the possibility of connections with a variety of areas of mathematics. The class of all graphs under subgraph inclusion provides a natural example. This may be studied using, among other things, modal logic - a logic which is interpreted on relations. The first-order modal logic of the graph theoretic potentialist system is rather power; for instance, one can express graph two-colourability, and even the finiteness and countability of graphs. The primary aim in this area would be to determine the precise strength of this logic. This project falls within the EPSRC Logic and Combinatorics research area.