The structural complexity of models of arithmetic

Abstract

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than ω and that non-standard models of true arithmetic must have Scott rank greater than ω. Other than that there are no restrictions. By giving a reduction via Δin1bi-interpretability from the class of linear orderings to the canonical structural ω-jump of models of an arbitrary completion T of PA we show that every countable ordinal α > ω is realized as the Scott rank of a model of T.