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

TL;DR

This paper investigates the logical strength of the Rival-Sands theorem in reverse mathematics, establishing equivalences with well-known subsystems and extending the theorem to new classes of partial orders.

Contribution

It characterizes the Rival-Sands theorem's strength for various classes of partial orders within reverse mathematics and extends the theorem to partial orders without infinite antichains.

Findings

Equivalence of Rival-Sands theorem for finite width partial orders to  + ext{ADS}

Equivalence for width  to ext{ADS}

Extension of Rival-Sands theorem to partial orders without infinite antichains, equivalent to arithmetical comprehension

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 $\mathsf{RCA}_0$.