A tame cantor set

Philipp Hieronymi

Research output: Contribution to journalArticlepeer-review


A Cantor set is a non-empty, compact subset of ℝ 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 model-theoretically tame Cantor set. This answers questions raised by Friedman, Kurdyka, Miller and Speissegger. Our work 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 have never been used in the setting of expansions of the real field.

Original languageEnglish (US)
Pages (from-to)2063-2104
Number of pages42
JournalJournal of the European Mathematical Society
Issue number9
StatePublished - 2018


  • Borel sets
  • Cantor set
  • Expansions of the real field
  • Monadic second-order theory of one successor
  • Quantifier elimination
  • Tame geometry

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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