Dynamic Set Theory
Yvette Oortwijn
Abstract:
This thesis develops a novel approach to a problem in the philosophy of mathematics, namely the problem of how to model the constantly evolving universe of mathematical objects. In the foundations of mathematics as formulated by set theory, this problem manifests itself as the question why the universe of sets is not a set itself, referred to as the why-question. Potentialism is a view in the philosophy of mathematics aiming to answer this question by interpreting the hierarchy of sets as not actually but potentially existing. This view has recently gained renewed interest from both philosophers and mathematicians. However, Potentialism as currently implemented is not able to do what it purports to do. In this thesis, a dynamic formulation of set theory (DST) will be developed in order to overcome this deficiency. Dynamic logic fits the philosophical view that Potentialism aims to capture since it can model the growth of information, which in this case refers to the expansion of the hierarchy of sets. Based on this dynamic formulation of set theory, a formal comparison between ZF and DST and a consideration on whether DST is able to answer the why-question is made. This formal comparison between ZF and DST shows that the two theories are interpretable in ZF given the right choice of semantics (either both classical, both intuitionistic or intuitionistic DST into classical ZF). However, DST is not able to give an in-depth answer to the why-question that is not open to existing criticism. The use of dynamic logic does allow for an independent motivation for this method of modelling the universe of sets. Moreover, it is argued that there are good reasons to think that a fully satisfactory, in-depth answer to the why-question might be too high an aim.