On the Consistency of Peano Arithmetic in a Proof-theoretic Semantics for Classical Logic

Abstract: We give a proof of the consistency of Peano Arithmetic (PA) within a novel semantic framework for classical logic due to Sandqvist. The argument proceeds by constructing an object $\mathfrak{A}$ -- the arithmetic base -- which supports all axioms of PA and can be shown to not support $\bot$, relative to a well-foundedness assumption equivalent to $\epsilon_0$-induction. This framework belongs to the paradigm of proof-theoretic semantics, that unlike model-theoretic approaches, offers a finitistically acceptable account in the spirit of Hilbert's Programme.