next up previous contents
Next: Introduction

Towards the Interpretability Logic of all Reasonable Arithmetical Theories

Joost J. Joosten

12 december 1998


The subject of this paper is the interpretability logic of all reasonable arithmetical theories, which we baptize $\mathit{GIL}$. This logic has long been conjectured by Albert Visser to be $\mathit{ILW}^*$. No modal completeness result was known for $\mathit{ILW}^*$. In order to provide a completeness proof of $\mathit{ILW}^*$, a good understanding of the sublogic $\mathit{ILM_0}$ seems indispensable. But there was no modal completeness result for $\mathit{ILM_0}$ either. In this paper a method for constructing models is developed. By this model-construction method we obtain a new proof of the modal completeness and decidability of $\mathit{ILM}$. Moreover do we obtain the modal completeness of $\mathit{ILM_0}$. Furthermore, a new principle, P0, is introduced and studied. This principle is seen to be arithmetically valid and is completely independent with regard to the other principles studied here. The new logic $\mathit{ILP_0}$ turns out to be modally incomplete. The conjecture of $\mathit{ILW}^*$ being the interpretability logic of all reasonable arithmetical theories is thus falsified.


Joost Joosten