The structural complexity of models of arithmetic

TL;DR

This paper investigates the possible Scott ranks of countable models of Peano arithmetic, establishing bounds for non-standard models and showing the realization of all ordinals above 6 as Scott ranks through a bi-interpretability reduction.

Contribution

It characterizes the Scott ranks of models of PA, proving bounds for non-standard models and demonstrating that all ordinals above 6 can be realized as Scott ranks.

Findings

No non-standard model has Scott rank less than 6.

Non-standard models of true arithmetic have Scott rank greater than 6.

Every countable ordinal greater than 6 can be realized as a Scott rank.

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 $\omega$ and that non-standard models of true arithmetic must have Scott rank greater than $\omega$. Other than that there are no restrictions. By giving a reduction via $\Delta^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $\alpha>\omega$ is realized as the Scott rank of a model of $T$.