AI 中文总结
本文证明强相对化 $\Delta^1_1$ 的实数是超低实数，还刻画了强相对化 $\Delta^0_\alpha$ 类的实数性质，得到相关基数结果并给出非平凡实例。
AI 中文摘要
对于实数 $f\in \mathbb{N}^\mathbb{N}$，通常并非每个同时为 $\Sigma^1_1(f)$ 和 $\Pi^1_1(f)$ 的集合都是 $\Delta^1_1$ 集合的 $f$ 截面。但存在部分实数 $f$ 满足该性质，我们称这类实数强相对化 $\Delta^1_1$。本文将证明，强相对化 $\Delta^1_1$ 的实数恰好是超低实数，即满足 $\omega_1^f=\omega_1^{\text{ck}}$ 的 $f\in \mathbb{N}^\mathbb{N}$。我们还研究强相对化类 $\Delta^0_\alpha$（$1\leq \alpha<\omega_1^{\text{ck}}$）的实数：将 $\mathbb{N}$ 的 $\Delta^0_1$ 子集强相对化的实数刻画为可计算受控实数；对 $1\leq \alpha<\omega_1^{\text{ck}}$，证明存在连续统多个强相对化 $\mathbb{N}$ 的 $\Delta^0_\alpha$ 子集的实数；还找到强相对化贝尔空间的 $\Delta^0_\alpha$（$1\leq \alpha<\omega_1^{\text{ck}}$）子集的非平凡实数例子。
英文摘要
For a real $f\in \mathbb{N}^\mathbb{N}$, it is in general not the case that every set which is both $Σ^1_1(f)$ and $Π^1_1(f)$ is the $f$-section of a $Δ^1_1$ set. However, there are reals $f$ for which this, in fact, does happen; we say that such a real strongly relativizes $Δ^1_1$. In this paper, we will prove that the reals which strongly relativize $Δ^1_1$ are exactly the hyperlow reals, i.e., the $f\in \mathbb{N}^\mathbb{N}$ with $ω_1^f=ω_1^{\text{ck}}$. We will also study reals that strongly relativize the classes $Δ^0_α$ for $1\leq α<ω_1^{\text{ck}}$. We characterize the reals that strongly relativize $Δ^0_1$ subsets of $\mathbb{N}$ as the reals which are computably dominated. For $1\leq α<ω_1^{\text{ck}}$, we show that there are continuum many reals which strongly relativize $Δ^0_α$ subsets of $\mathbb{N}$. We will also find non-trivial examples of reals which strongly relativize $Δ^0_α$ ($1\leq α<ω_1^{\text{ck}}$) subsets of Baire space.
Comments19 pages
