The uniform content of partial and linear orders

TL;DR

This paper compares various principles related to linear and partial orders, showing strict reducibility hierarchies among them using Weihrauch and computable reducibility, and introduces new stable variants.

Contribution

It introduces the principle ADC and its variants, establishing their relative strength compared to ADS and SADS, and analyzes the reducibility relations among stable variants of CAC.

Findings

ADC is strictly weaker than ADS under Weihrauch reducibility.

SADS is not Weihrauch reducible to ADC.

SCAC is strictly weaker than WSCAC under computable reducibility.

Abstract

The principle $ADS$ asserts that every linear order on $\omega$ has an infinite ascending or descending sequence. This has been studied extensively in the reverse mathematics literature, beginning with the work of Hirschfeldt and Shore. We introduce the principle $ADC$, which asserts that linear order has an infinite ascending or descending chain. The two are easily seen to be equivalent over the base system $RCA_0$ of second order arithmetic; they are even computably equivalent. However, we prove that $ADC$ is strictly weaker than $ADS$ under Weihrauch (uniform) reducibility. In fact, we show that even the principle $SADS$, which is the restriction of $ADS$ to linear orders of type $\omega + \omega^*$, is not Weihrauch reducible to $ADC$. In this connection, we define a more natural stable form of $ADS$ that we call $General\text-SADS$, which is the restriction of $ADS$ to linear orders of type $k + \omega$, $\omega + \omega^*$, or $\omega + k$, where $k$ is a finite number. We define $GeneralSADC$ analogously. We prove that $GeneralSADC$ is not Weihrauch reducible to $SADS$, and so in particular, each of $SADS$ and $SADC$ is strictly weaker under Weihrauch reducibility than its general version. Finally, we turn to the principle $CAC$, which asserts that every partial order on $\omega$ has an infinite chain or antichain. This has two previously studied stable variants, $SCAC$ and $WSCAC$, which were introduced by Hirschfeldt and Jockusch, and by Jockusch, Kastermans, Lempp, Lerman, and Solomon, respectively, and which are known to be equivalent over $RCA_0$. Here, we show that $SCAC$ is strictly weaker than $WSCAC$ under even computable reducibility.