与升序/降序序列原理的（非凡）等价性
(Extra)ordinary equivalences with the ascending/descending sequence principle
AI总结：
本文在逆数学框架下证明Rival-Sands定理的不同版本分别等价于IΣ⁰₂+ADS、ADS和SADS，并将其推广到无无限反链的偏序，首次给出由这些原理精确刻画的数学定理实例。
摘要：
我们以逆数学的风格分析如下由Rival和Sands提出的定理的公理强度：“每个有限宽度的无限偏序$P$都包含一个无限链$C$，使得$P$中的每个元素要么与$C$中没有任何元素可比，要么与$C$中无穷多个元素可比。”我们的主要结果如下。任意有限宽度的无限偏序的Rival-Sands定理在$\mathsf{RCA}_0$上等价于$\mathsf{I}Σ^0_2 + \mathsf{ADS}$。对每个固定的$k \geq 3$，宽度$\leq\\! k$的无限偏序的Rival-Sands定理在$\mathsf{RCA}_0$上等价于$\mathsf{ADS}$。可分解为两条链的并的无限偏序的Rival-Sands定理在$\mathsf{RCA}_0$上等价于$\mathsf{SADS}$。这里$\mathsf{RCA}_0$表示递归理解公理系统，$\mathsf{I}Σ^0_2$表示$Σ^0_2$归纳模式，$\mathsf{ADS}$表示升序/降序序列原理，$\mathsf{SADS}$表示稳定升序/降序序列原理。据我们所知，这些偏序版本的Rival-Sands定理是一般数学文献中第一批其强度被$\mathsf{I}Σ^0_2 + \mathsf{ADS}$、$\mathsf{ADS}$和$\mathsf{SADS}$精确刻画的定理的例子。此外，我们通过将Rival-Sands定理推广到没有无限反链的无限偏序，给出了一个新的纯组合结果，并且我们证明这一推广在$\mathsf{RCA}_0$上等价于算术理解。
英文摘要：
We analyze the axiomatic strength of the following theorem due to Rival and Sands in the style of reverse mathematics. "Every infinite partial order $P$ of finite width contains an infinite chain $C$ such that every element of $P$ is either comparable with no element of $C$ or with infinitely many elements of $C$." Our main results are the following. The Rival-Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf{I}Σ^0_2 + \mathsf{ADS}$ over $\mathsf{RCA}_0$. For each fixed $k \geq 3$, the Rival-Sands theorem for infinite partial orders of width $\leq\! k$ is equivalent to $\mathsf{ADS}$ over $\mathsf{RCA}_0$. The Rival-Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf{SADS}$ over $\mathsf{RCA}_0$. Here $\mathsf{RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf{I}Σ^0_2$ denotes the $Σ^0_2$ induction scheme, $\mathsf{ADS}$ denotes the ascending/descending sequence principle, and $\mathsf{SADS}$ denotes the stable ascending/descending sequence principle. To our knowledge, these versions of the Rival-Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf{I}Σ^0_2 + \mathsf{ADS}$, by $\mathsf{ADS}$, and by $\mathsf{SADS}$. Furthermore, we give a new purely combinatorial result by extending the Rival-Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf{RCA}_0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
