Numeral completeness of weak theories of arithmetic

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 Godel'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.

Automated summary

We study the numeral forms of completeness and consistency for weak theories, including $\mathsf {S}<^>1_2$ and $\mathsf {EA}$. This exploration leads us to examine the derivability conditions required to establish these results, as well as present a weak form of Godel's Second Incompleteness Theorem without using 'provability implies provable provability'. Additionally, we introduce a provability predicate that satisfies the mentioned derivability condition for weak theories, and present a completeness result through consistency statements. Moreover, we provide characterizations of the provability predicates for which the numeral results hold, with $\mathsf {EA}$ as the surrounding theory, and offer results on functions that compute finitist consistency statements.