Natural Axiomatic Theories and Consistency Strength: A Lakatosian Approach to the Linearity Conjecture
Lide Grotenhuis
Abstract:
In the literature on relative consistency results, one often encounters the claim that all natural axiomatic theories are linearly ordered in terms of consistency strength. Without a precise definition of a natural theory, it is not clear how to assess the truth of this claim or how to judge whether the known instances of nonlinearity constitute genuine counterexamples to it. The general aim of this thesis is to take a first step towards such a precise definition. To this end, the thesis consists of two parts. First, after arguing that a pursuit of such a definition is worthwhile, I develop a method for working towards such a definition. This method is primarily inspired by Lakatosâ€™ approach to mathematical concept-formation, whose main tenet is that mathematical concepts develop in response to the emergence of counterexamples. Second, I apply the method and analyze the known instances of nonlinearity, including those recently suggested by Hamkins. Building on this analysis, I develop the following tentative definition: an axiomatic theory is natural if its axioms do not carry meta-information and if the theory is presented in a surveyable manner.