The ordering principle states that every finite linear order has a least element. We show that, in the relativized setting, the surjective weak pigeonhole principle for polynomial time functions does not prove a Herbrandized version of the ordering principle over T-2(1). This answers an open question raised in Buss et al. [2012] and completes their program to compare the strength of Jerabek's bounded arithmetic theory for approximate counting with weakened versions of it.

Peer Reviewed