(Extra)ordinary equivalences with the ascending/descending sequence principle

Abstract

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 $\mathsf{I}\Sigma^0_2 + \mathsf{ADS}$ 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, $\mathsf{I}\Sigma^0_2$ denotes the $\Sigma^0_2$ 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 $\mathsf{I}\Sigma^0_2 + \mathsf{ADS}$, 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 $\mathsf{RCA}_0$.