Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
复制标题
在射影层次的给定级别上具有可数横截面的非均匀集
作者：
V. Kanovei;V. Lyubetsky
我们提出一个集合论模型，在其中，对于给定的\(n\geq2\)，在\(\mathbb{R}\times\mathbb{R}\)中存在一个非相对序数可定义（ROD）-可一致化的平面浅体\(\varPi^1_n\)集合，它的所有垂直截面都是可数集（实际上是维塔利类），然而所有具有可数截面的平面黑体\(\bf{\Sigma}^1_n\)集合都是\(\bf{\Delta}^1_{n + 1}\)-可一致化的。因此在这个模型中，对于具有可数截面的集合的相对序数可定义（ROD）-一致化原理恰好在给定的射影层次首次失效。
We present a model of set theory, in which, for a given $nge2$, there exists a non-ROD-uniformizable planar lightface $varPi^1_n$ set in $mathbb R imesmathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $fSigma^1_n$ sets with countable cross-sections are $fDelta^1_{n+1}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level.