Numeral completeness of weak theories of arithmetic

Abstract

We study numeral forms of completeness and consistency for $\mathsf {S}^1_2$ and other weak theories, like $\mathsf {EA}$. This gives rise to an exploration of the derivability conditions needed to establish the mentioned results; a presentation of a weak form of Gödel’s Second Incompleteness Theorem without using ‘provability implies provable provability’; a provability predicate that satisfies the mentioned derivability condition for weak theories; and a completeness result via consistency statements. Moreover, the paper includes characterizations of the provability predicates for which the numeral results hold, having $\mathsf {EA}$ as the surrounding theory, and results on functions that compute finitist consistency statements.