TY - JOUR
T1 - A tame cantor set
AU - Hieronymi, Philipp
N1 - Funding Information:
I thank Chris Miller for bringing this question to my attention; I would also like to thank Anush Tserunyan for answering my questions about descriptive set theory, Lou van den Dries for answering questions about o-minimality, and Carl Jockusch and Paul Schupp for answering questions about the monadic second-order theory of one successor. The author was partially supported by NSF grant DMS-1300402 and by UIUC Campus Research Board award 14194.
Funding Information:
The author was partially supported by NSF grant DMS-1300402 and by UIUC Campus Research Board award 14194.
Publisher Copyright:
© European Mathematical Society 2018.
PY - 2018
Y1 - 2018
N2 - 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.
AB - 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.
KW - Borel sets
KW - Cantor set
KW - Expansions of the real field
KW - Monadic second-order theory of one successor
KW - Quantifier elimination
KW - Tame geometry
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U2 - 10.4171/JEMS/806
DO - 10.4171/JEMS/806
M3 - Article
AN - SCOPUS:85052019702
SN - 1435-9855
VL - 20
SP - 2063
EP - 2104
JO - Journal of the European Mathematical Society
JF - Journal of the European Mathematical Society
IS - 9
ER -