常微分方程与描述集合论：一维Cauchy问题解的唯一性与整体性
Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension
摘要：
我们研究了一些自然集所产生的理论中的常微分方程在一个变量的观点，描述集合论，特别是将它们分类内的博雷尔层次。证明了有唯一解的常微分方程的柯西问题集是$Π^0_2$ -完备的，局部有唯一解的柯西问题集是$Σ^0_3$ -完备的.证明了具有整体解的Cauchy问题集是$Σ^0_4$ -完备的，而对于每个初始条件都具有整体解的常微分方程集是$Π^0_3$ -完备的.证明了满足唯一性和整体性的柯西问题集是$Π^0_2$ -完备的。
英文摘要：
We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $Π^0_2$-complete and that the set of Cauchy problems which locally have a unique solution is $Σ^0_3$-complete. We prove that the set of Cauchy problems which have a global solution is $Σ^0_4$-complete and that the set of ordinary differential equation which have a global solution for every initial condition is $Π^0_3$-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is $Π^0_2$-complete.
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