Near coherence of filters. I: Cofinal equivalence of models of arithmetic (Q1090677)

Two models of full arithmetic are cofinally equivalent (c.e.) if they both can be cofinally embedded in one and the same model. Firstly, any two tall models of the same cofinality are c.e. The matter is much more delicate with short models, in that for minimal non-standard (n.s.) models, c.e. coincides with isomorphism. The principle of near coherence of filters (NCF) is investigated in this regard. It means that any two short n.s. models are c.e. NCF comes out to be equivalent to various properties of free ultrafilters; it is consistent with (and independent from) ZFC (Shelah). In particular NCF has some import on the dominating number d, i.e. the smallest cardinality of a dominating family of functions. E.g., NCF implies that all short n.s. models have cofinality d, and that any fewer than d short n.s. models have isomorphic cofinal submodels.