多值函数的秩不连续性
A Rank for Discontinuities of Multivalued Functions
AI总结：
研究多值函数的不连续性，通过序数排名和Wadge可约性游戏，定义RT^n_{α,β}并计算其不连续性秩，展示其与ACC_N的区别。
AI中文摘要：
我们通过为多值函数（也称为问题）$P$定义域中没有局部实现者的点赋予一个序数秩，来研究贝尔空间上多值函数的不连续性。对于每个可数序数$\alpha$，令$\mathsf{ACC}_{\mathbb N}^{\alpha}$为在至多$\alpha$次尝试内解决$\mathsf{ACC}_{\mathbb N}$的问题，每次尝试会提供一个新实例。主要定理表明，对于任何问题$P$，以下条件等价：(i) $P$在某个集合上不连续，该集合中所有点的康托尔 - 本迪克森秩至多为$\alpha$；(ii) $\mathsf{ACC}_{\mathbb N}^{\alpha}\leq_{\mathrm W}^{*}P$。我们还通过$P$的瓦吉风格不连续性博弈来刻画这些性质。将其应用于薄集和无色拉姆齐定理，扩展$\mathsf{RT}^{n}_{k,j}$到序数参数，定义$\mathsf{RT}^{n}_{\alpha,\beta}$并计算其不连续秩。$\mathsf{RT}^n_{\alpha,\beta}$给出了具有每个可数秩不连续性且不可约化为$\mathsf{ACC}_{\mathbb{N}}$的问题示例，通过识别错误的可猜测性概念实现了与$\mathsf{ACC}_{\mathbb{N}}$的分离。
英文摘要：
We study discontinuity of multivalued functions (also known as problems) $P$ on Baire space by assigning an ordinal rank to points in the domain of $P$ that have no local realizers. For each countable ordinal $α$, let $\mathsf{ACC}_{\mathbb N}^α$ be the problem of solving $\mathsf{ACC}_{\mathbb N}$ in at most $α$ many attempts, with a new instance provided for each attempt. Our main theorem shows that, for any problem $P$, the following are equivalent: (i) $P$ is discontinuous on some set all of whose points have Cantor--Bendixson rank at most $α$, and (ii) $\mathsf{ACC}_{\mathbb N}^α\leq_{\mathrm W}^{*}P$. This extends to points: $P \geq_{\mathrm{W}}^* \mathsf{ACC}_{\mathbb{N}}^α$ via a forward function that sends $\#^{\mathbb{N}}$ to $p \in \operatorname{dom}(P)$ if and only if $P$ is discontinuous on a set $A$ with $\operatorname{rank}_A(p) \leq α$ and $P$ has no continuous realizer for any neighborhood of $p$. We also characterize these properties via a Wadge-style discontinuity game for $P$. We apply this framework to the thin set and achromatic Ramsey theorems. Extending $\mathsf{RT}^{n}_{k,j}$ to ordinal parameters, we define $\mathsf{RT}^{n}_{α,β}$ and compute their ranks of discontinuity. The problems $\mathsf{RT}^n_{α,β}$ provide examples of problems with discontinuities of each countable rank which are not reducible to $\mathsf{ACC}_{\mathbb{N}}$. The separation from $\mathsf{ACC}_{\mathbb{N}}$ is obtained via the notion of guessability with identified errors.
