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

Abstract

We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] 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 IΣ 0 2 + ADS over RCA0. For each fixed k ≥ 3, the Rival-Sands theorem for infinite partial orders of width ≤ k is equivalent to ADS over RCA0. The Rival-Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to SADS over RCA0. Here RCA0 denotes the recursive comprehension axiomatic system, IΣ 0 2 denotes the Σ 0 2 induction scheme, ADS denotes the ascending/descending sequence principle, and 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 IΣ 0 2 + ADS, by ADS, and by 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 RCA0.