Potentialist Systems for Set Theory and Beyond

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.