The uniform content of partial and linear orders

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.