论文标题
实际数字上的结构连接性：O最小性和不可证明性
Connectedness in structures on the real numbers: o-minimality and undecidability
论文作者
论文摘要
我们启动对具有可定义集合的路径组件的属性的实数集的结构的调查。 $（\ Mathbb {r}，<）$上的所有O \ nobreakdash- \ hspace {0pt}最小结构都具有属性，$（\ mathbb {r}，+，\ cdot，\ cdot，\ mathbb {n}）的所有扩展都一样。我们的主要分析几何结果是，通过布尔值组合（任何Arities）的布尔组合$ \ nobreakdash- \ hspace {0pt {0pt} mathbb {r}，<，+）$的任何此类扩展，或者定义了$（\ mathbb n，+notimph）$ \ nobreakdash- \ hspace {0pt} $（我们还表明，通过$ \ mathbb {n}^n $的子集的$（\ mathbb {r}，<， +，\ mathbb {n}）$的任何给定扩展。通过考虑连接的组件或准综合体而不是路径组件来产生变化。
We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\nobreakdash-\hspace{0pt}minimal structures on $(\mathbb{R},<)$ have the property, as do all expansions of $(\mathbb{R},+,\cdot,\mathbb{N})$. Our main analytic-geometric result is that any such expansion of $(\mathbb{R},<,+)$ by boolean combinations of open sets (of any arities) either is o\nobreakdash-\hspace{0pt}minimal or defines an isomorph of $(\mathbb N,+,\cdot\,)$. We also show that any given expansion of $(\mathbb{R}, <, +,\mathbb{N})$ by subsets of $\mathbb{N}^n$ ($n$ allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.
