论文标题
可计数排名在第一和第二个投影级别
Countable ranks at the first and second projective levels
论文作者
论文摘要
等级是描述性集理论中的一个概念，它描述了诸如波兰空间的封闭子集的cantor-bendixson等级，在$ c [0,1] $中的一组可区分功能的封闭子集上，例如kechris-woodin等级，例如kechris-woodin等级以及在描述性集合理论和真实分析中的许多其他等级。许多自然等级的复杂性为$π^1_1 $或$σ^1_2 $。我们建议将集合中最小的排名理解为衡量其复杂性的量度。因此，目的是了解此类等级可能具有哪些长度。主要结果决定了在第一和第二个投影水平上可数等级长度的上流。此外，我们表征了$σ^1_2 $集的特定类别上的可数等级的存在。一方面具有可数等级的$σ^1_2 $设置的连接和$σ^1_2 $ borel set的另一个导致猜想，该猜想以描述性的理论（例如Mansfield-Solovay Theorem和Kanovei和Lyubetsky的最新结果）统一了几个结果。
A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the Kechris-Woodin rank and many other ranks in descriptive set theory and real analysis. The complexity of many natural ranks is $Π^1_1$ or $Σ^1_2$. We propose to understand the least length of ranks on a set as a measure of its complexity. Therefore, the aim is to understand which lengths such ranks may have. The main result determines the suprema of lengths of countable ranks at the first and second projective levels. Furthermore, we characterise the existence of countable ranks on specific classes of $Σ^1_2$ sets. The connections arising between $Σ^1_2$ sets with countable ranks on the one hand and $Σ^1_2$ Borel sets on the other lead to a conjecture that unifies several results in descriptive set theory such as the Mansfield-Solovay theorem and a recent result of Kanovei and Lyubetsky.
