论文标题
分析层次结构中的罗杰斯半纹身：有限家庭的情况
Rogers semilattices in the analytical hierarchy: The case of finite families
论文作者
论文摘要
可数家族$ s $的编号是从自然数$ω$集中到$ s $的汇总地图。论文研究Rogers的半纹身，即由$ s \ subset p（ω）$的家庭之间的降低性诱导的上半纹身。在集合理论ZF+DC+PD中，我们获得了来自分析层次结构各个级别的家族的以下结果。 对于非零数字$ n $，如果$ n $为奇数，则$ e^1_n $我们表示$π^1_n $，而$σ^1_n $如果$ n $甚至是。我们表明，对于有限的family $ s $ e^1_n $ sets，其rogers $ e^1_n $ -semilattice具有最大的元素，并且仅当$ s $包含设置理论包容的最低元素。此外，如果$ s $没有$ \ subseteq $ - 易碎元素，则相应的rogers $ e^1_n $ -semilattice向上密集。
A numbering of a countable family $S$ is a surjective map from the set of natural numbers $ω$ onto $S$. The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings, for families $S\subset P(ω)$. Working in set theory ZF+DC+PD, we obtain the following results on families from various levels of the analytical hierarchy. For a non-zero number $n$, by $E^1_n$ we denote $Π^1_n$ if $n$ is odd, and $Σ^1_n$ if $n$ is even. We show that for a finite family $S$ of $E^1_n$ sets, its Rogers $E^1_n$-semilattice has the greatest element if and only if $S$ contains the least element under set-theoretic inclusion. Furthermore, if $S$ does not have the $\subseteq$-least element, then the corresponding Rogers $E^1_n$-semilattice is upwards dense.
