驯服的康托尔集

摘要：

一个康托尔集是一个非空的紧集，它既没有内部点也没有孤立点。本文构造了一个康托尔集$K\subseteq \mathbb{R}$，使得在$(\mathbb{R},<,+,\cdot,K)$中可定义的所有集合都是Borel的.此外，我们证明了$(\mathbb{R},<,+,\cdot,K)$的量词消除和完备性结果，使集合$K$成为理论上驯服的康托尔集的第一个例子。这就回答了弗里德曼、库尔迪卡、米勒和斯佩塞格提出的问题。本文的工作主要依赖于关于无限词上自动机的结果，特别是著名的Büchi关于一个后继者的一元二阶理论的定理和关于Muller自动机的McNaughton定理，这些定理从未在实域的展开式中使用过。

英文摘要：

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 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.

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