论文标题
$ \boldsymbolπ_1^1 \ mathsf {-ca} _0 $的Weihrauch晶格
The Weihrauch lattice at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$: the Cantor-Bendixson theorem
论文作者
论文摘要
本文继续通过Weihrauch降低性框架连接反向数学和可计算分析的程序。特别是，我们考虑了与波兰空间的完美子集有关的问题，研究完美的定理，康托克森定理以及它们引起的各种问题。在反向数学的框架中，这些定理分别等同于$ \ m athsf {atr} _0 $和$ \boldsymbolπ_1^1 \ mathsf {-ca} _0 $，这是两个最强的二阶算术子系统，这是所谓的两个算术中的两个最强子系统。据我们所知，这是对$ \boldsymbolπ_1^1 \ mathsf {-ca} _0 $在weihrauch lattice中的第一次系统研究。 我们表明，我们研究的某些问题的强度取决于正在考虑的波兰空间的拓扑特性，而一旦空间足够丰富，其他问题的强度具有相同的强度。
This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor-Bendixson theorem and various problems arising from them. In the framework of reverse mathematics these theorems are equivalent respectively to $\mathsf{ATR}_0$ and $\boldsymbolΠ_1^1\mathsf{-CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.
