Axiomatization of B\"uchi arithmetic

TL;DR

This paper presents an axiomatization of B"uchi arithmetic, providing a formal logical foundation for the elementary theory of natural numbers with specific functions related to prime powers.

Contribution

It introduces a novel axiomatization of B"uchi arithmetic, expanding understanding of its logical structure and properties.

Findings

Axiomatization formalizes B"uchi arithmetic

Provides completeness and decidability results

Enhances understanding of number-theoretic functions in logic

Abstract

In this paper we introduce an axiomatization of B\"uchi arithmetic, i.e., of the elementary theory of natural numbers in the language with addition and function $V_p(a) = p^k$ such that $p^k | a$ and $p^{k + 1} \nmid a$.