论文标题
完全确定的Borel集和可衡量性
Completely determined Borel sets and measurability
论文作者
论文摘要
我们考虑说明$ \ mathsf {C \ text-dm} $的反向数学强度：“每个完全确定的borel集都可以测量。”超过$ \ mathsf {wwkl} _0 $，我们获得以下结果类似于先前研究的类别案例。 $ \ mathsf {c \ text-dm} $严格介于$ \ mathsf {atr} _0 $和$ \ mathsf {l} _ {ω__1，ω} \ text- \ text- \ mathsf {ca} $之间。每当$ \ mathsf {c \ text-dm} $的$ω$ -MODEL的二阶部分时，每当$ω$ -MODEL的二阶部分，那么对于M $中的每个$ z \，M $中的每个$ r \ in m $中，$ r $ as $ r $ is $ ru $ is $ us $δ^1_1 $δ1_1$ -random to to $ z $。另一方面，如果没有$ \ MATHSF {WWKL} _0 $，所有集合的测量均为零（按照$ \ Mathsf {C \ Text-DM} $测量，它可以真实地遵循$ \ neg \ negsf $ \ mathsf {rca} _0 $。
We consider the reverse math strength of the statement $\mathsf{C\text-DM}$:"Every completely determined Borel set is measurable." Over $\mathsf{WWKL}_0$, we obtain the following results analogous to the previously studied category case. $\mathsf{C\text-DM}$ lies strictly between $\mathsf{ATR}_0$ and $\mathsf{L}_{ω_1,ω}\text-\mathsf{CA}$. Whenever $M\subseteq 2^ω$ is the second-order part of an $ω$-model of $\mathsf{C\text-DM}$, then for every $Z \in M$, there is a $R \in M$ such that $R$ is $Δ^1_1$-random relative to $Z$. On the other hand, without $\mathsf{WWKL}_0$, all sets have measure zero (as measured according to $\mathsf{C\text-DM}$), and it follows vacuously that $\neg \mathsf{WWKL}_0$ implies $\mathsf{C\text-DM}$ over $\mathsf{RCA}_0$.
