Fibonacci Numbers and Model-Complete Axiomatization of Presburger Arithmetic Expanded with a Beatty Sequence

TL;DR

This paper presents a complete axiomatization of a structure involving integers, order, addition, and a function based on the golden ratio, proving model-completeness with an expanded language.

Contribution

It introduces a recursive theory that fully axiomatizes the structure with a Beatty sequence function and establishes its model-completeness.

Findings

A recursive axiomatization of the structure involving the Fibonacci sequence.

Proof of model-completeness in an expanded language.

Characterization of the structure's logical properties.

Abstract

We introduce a recursive theory that completely axiomatizes the structure $\langle \mathbb{Z},<, +,f,0\rangle$ where $f$ is the function that maps each $x$ to the integer part of $\varphi x $, with $\varphi$ the golden ratio. We prove that our axiomatization is model-complete in a language expanded with a function which we which we refer as the Fibonacci floor function.