2015/07/12 by Philipp Hieronymi, Hieronymi, Philipp
Mathematics · #03D05 #03E15 #28E15 #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #Primary 03C64 #Secondary 03C10 #math.LO #msc:03C10 #msc:03C64 #msc:03D05 #msc:03E15 #msc:28E15
paper · pdf · doi:10.48550/arxiv.1507.03201
openalex publication_date 2015/07/12 · arxiv created 2016/05/03 · arxiv updated 2016/05/04 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set K⊆ ℝ is constructed such that every set definable in (ℝ,<,+,⋅,K) is Borel. In addition, we prove quantifier-elimination and completeness results for (ℝ,<,+,⋅,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 Büchi'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.