A tame Cantor set

TL;DR

This paper constructs a special Cantor set within the real numbers that ensures all definable sets are Borel, and proves model-theoretic properties like quantifier-elimination, making it the first tame Cantor set with such features.

Contribution

It introduces the first example of a model-theoretically tame Cantor set with quantifier-elimination and completeness in the structure $( eal, <, +, imes, K)$.

Findings

All definable sets in the structure are Borel.

The structure admits quantifier-elimination and is complete.

This is the first example of a tame Cantor set with these properties.

Abstract

A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we prove quantifier-elimination and completeness results for $(\mathbb{R},<,+,\cdot,K)$, making the set $K$ the first example of a modeltheoretically tame Cantor set. This answers questions raised by Friedman, Kurdyka, Miller and Speissegger. The work in this paper depends crucially on results about automata on infinite words, in particular B\"uchi's celebrated theorem on the monadic second-order theory of one successor and McNaughton's theorem on Muller automata, which had never been used in the setting of expansions of the real field.