Branchwise-real trees and bisimulations of potentialist systems

Thesis

This thesis concerns two topics. The first treats R-trees, which are a certain kind of metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of partial orders underlying R-trees be characterised by the fact that every branch is order-isomorphic to a real interval? I first answer this question in the negative, then go on to establish a connection between these trees and traditional set-theoretic trees. This connection is then put to work, answering refinements of Favre and Jonsson's question, yielding several independence results. I next move on to consider the existence of examples of these partial orders without non-trivial automorphisms. I provide constructions of these subject to increasingly strong uniformity conditions. While these constructions all take place in ZFC, they have a strong forcing flavour.

The second topic deals with bisimulations of potentialist systems, which are first-order Kripke models based on embeddings. Given a first-order theory T we can impose a potentialist structure on the class of models of T by taking either all embeddings or all substructure inclusions between models. I show that these two systems are always bisimilar. Next, by connecting with a generalisation of the Ehrenfeucht-Fraïsé game, I show the equivalence of the existence of a bisimulation with elementary equivalence with respect to an infinitary language. Finally, I turn to the question of when a class-sized potentialist system is bisimilar to a set-sized one, providing two different sufficient conditions.

Authors: Adam-Day, S

• DOI: 10.5287/ora-zbpy742kq
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: forcing; bisimulation; modal model theory; Martin’s Axiom; Ehrenfeucht-Fraïssé game; separable; rigid; first-order Kripke model; uniform; branchwise-real tree; continuous grading; tree automorphism; road space; Suslin tree; potentialist system; independence result; countable chain condition
• Subjects: Forcing (Model theory); Combinatorial set theory; Modality (Logic)