A new principle in the interpretability logic of all reasonable arithmetical theories
Abstract
The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper from 2011 revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by ${\textbf{IL}}({\rm All})$. In this paper we present a new principle $\sf R$ in ${\textbf{IL}}({\rm All})$. We show that $\sf R$ does not follow from the logic ${\textbf{IL}}{\sf P_0W^*}$ that contains all previously known principles. This is done by providing a modal incompleteness proof of ${\textbf{IL}}{\sf P_0W^*}$: showing that $\sf R$ follows semantically but not syntactically from ${\textbf{IL}}{\sf P_0W^*}$. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with socalled Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle $\sf R$ is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.06902
 Bibcode:
 2020arXiv200406902G
 Keywords:

 Mathematics  Logic
 EPrint:
 Logic Journal of the Interest Group in Pure and Applied Logic 19 (1), 119, 2011