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Preprint Number 889

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889. Philipp Hieronymi
A tame Cantor set
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Submission date: 12 July 2015.

Abstract:

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.

Mathematics Subject Classification: Primary 03C64, Secondary 03C10, 03D05, 03E15, 28E15

Keywords and phrases:

Full text arXiv 1507.03201: pdf, ps.


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