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Potentialist Systems for Set Theory and Beyond

集合论及其他领域的势论系统

基本信息

批准号：
2271793
负责人：
金额：
--
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2271793
关键词：
Potentialist Systems Set Theory Beyond

项目摘要

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.

强迫是现代集合论中最重要的技术之一。它提供了一种从旧的集合论生成新模型的高度通用的方法。通常，强迫被用作获得一致性和独立性结果的工具；然而，它本身也是一个具有数学意义的对象。任何特定的强迫概念都会在集合论的模型类上诱导出一种关系，如果一个模型是另一个模型的强迫扩张，则两个模型是相关的。Andrzej Mostowski证明了在强意义下，任何有限偏序集都可以嵌入到模型M的一般多重宇宙中：由那些通过强迫-扩张关系从M可达的模型组成的子结构。乔尔·哈姆金斯使用了一种名为“集合论区块链”的新技术，他最近加强了这一结果，将一大类无限偏序集包括在内，并建立了许多其他的结构性质。事实上，M的一般多重宇宙类似于图灵度的结构，它也具有偏序集嵌入性质。这项研究提案的目的之一是调查这种联系会走多远。例如，一般的多重宇宙是否表现出嵌入-扩展性质：子偏序集的任何嵌入都可以扩展到整个偏序集的嵌入？强迫-扩展关系是位势系统的一个例子：一些理论的一类模型和与这些模型相关的扩展概念。拟议的研究方案还将调查这一更一般的领域。可以在集合论模型的类别上强加进一步的扩张概念，例如末端扩张的概念，并且也可以考虑其他自然的模型类别。后者的一个卓有成效的例子是所有算术模型的类，许多可拓概念都适用于它，并与集合论的情况密切相关。W.Hugh Woodin最近发现了通用的算术算法。这个算法列举了一个有限序列，并且有一个显著的性质：如果它在算术模型M中列举S，并且t是一个扩张S的有限序列，则存在M的一个末端扩张，其中它列举t。最近，Joel Hamkins对于集合论的可数模型发现了一个类似的结果：在右端扩张中有一个由E_2公式指定的有限集合，它可以包括任何期望的集合。这两个定理揭示了公式和模型之间深刻的新的相互作用，并为一系列应用打开了大门。在算术方面，一个推论是，在每一个算术模型中，都有一个丢番图方程，它没有解，但在某种程度上得到了解。在集合论方面，平行的结果是不存在具有最大E2图的集合论的可数模型。这方面的研究方向很多。通过取消模型是可计数的要求，结果能得到加强吗？是否存在其他类型的通用算法和通用有限集，例如，关于强制扩展的通用集？各种其他理论提供了电位论系统的有趣例子，以及与各种数学领域联系的可能性。子图包含下的所有图的类提供了一个自然的例子。除其他外，可以使用模态逻辑来研究这一点--一种对关系进行解释的逻辑。图论势能者系统的一阶模态逻辑是相当强的，例如，可以表示图的双色性，甚至图的有限性和可数性。这一领域的主要目标将是确定这一逻辑的确切强度。该项目属于EPSRC逻辑和组合学研究领域。