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.  and completes their program to compare the strength of Jerabek's bounded arithmetic theory for approximate counting with weakened versions of it.
CitationAtserias, A.; Thapen, N. The ordering principle in a fragment of approximate counting. "ACM transactions on computational logic", 2014, vol. 15, núm. 4.
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