Friedman-reflexivity: interpreters as consistoids

Abstract

Harvey Friedman shows that, over Peano Arithmetic, the consistency statement for a finitely axiomatised theory $A$ can be characterised as the weakest statement $C$ over Peano Arithmetic such that ${\sf PA}+C$ interprets $A$. We study which base theories $U$ have the property that, for any finitely axiomatised $A$, there is a weakest $C$ such that $U+C$ interprets $A$. We call such theories Friedman-reflexive. We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do not get the usual consistency statements here, but bounded, cut-free or Herbrand consistency statements. We prove a characterisation theorem for Friedman-reflexive sequential theories. We provide an example of a Friedman-reflexive sequential theory that substantially differs from the paradigm cases of Peano Arithmetic and Peano Corto. The consistency-like statements provided by a Friedman-reflexive base $U$ can be used to define a provability-like notion for a finitely axiomatised $A$ that interprets $U$ via an interpretation $K$ of $U$ in $A$. We call the modal logics based on this idea \emph{interpreter logics}. These logics satisfy the L\"ob Conditions. We provide conditions for when these logics extend {\sf S}4, {\sf K}45, and L\"ob's Logic. We show that, if either $U$ or $A$ is sequential, then the condition for extending L\"ob's Logic is fulfilled. Moreover, if our base theory $U$ is sequential and if, in addition, its interpreters can be effectively found, we prove Solovay's Theorem. This holds even if the provability-like operator is not necessarily representable by a predicate of G\"odel numbers. At the end of the paper, we briefly discuss how successful the coordinate-free approach is.