Near coherence of filters. I. Cofinal equivalence of models of arithmetic.

Abstract

We define cofinal equivalence to be the smallest equivalence relation on models of arithmetic such that every model is equivalent to all of its cofinal submodels. It is easy to classify tall models (those with no last sky) up to cofinal equivalence, but an attempt to do the same for short models leads to questions independent of the axioms of set theory. We introduce the set-theoretic principle (NCF) of near coherence of filters, whose effect is to make all short nonstandard models of arithmetic cofinally equivalent. We give several equivalent formulations and several consequences of NCF.