Categorical structure in coherent theory of arithmetic

Abstract

In this paper we provide a semantic and syntactic analysis of parametrised natural numbers object in coherent categories, or pr-coherent categories. Semantically, we show the definable functions in the initial pr-coherent category are exactly given by primitive recursive functions. We also show that any pr-coherent category supports the construction of bounded universal quantifications, which are absent in an arbitrary coherent category. Under these semantic consideration, we construct a coherent theory of arithmetic and we show its syntactic category is equivalent to the initial pr-coherent category. From a logical perspective, we also show that this theory can be identified as the {\Sigma}1-fragment of I{\Sigma}1. Thus as an application, we provide a structural proof of the classical result in proof theory that the strongly {\Sigma}1-representable functions in I{\Sigma}1 are exactly primitive recursive functions.