Neutrally Expandable Models of Arithmetic

TL;DR

This paper investigates the existence of neutral subsets in models of Peano Arithmetic, revealing conditions under which models are neutrally expandable and showing neutrality is not first-order definable.

Contribution

It introduces the concept of neutral sets and models, establishing new results on their existence in various models of PA and analyzing their properties.

Findings

Cofinal extensions of prime models are neutrally expandable.

Existence of ω₁-like neutrally expandable models.

No recursively saturated model is neutrally expandable.

Abstract

A subset of a model of ${\sf PA}$ is called neutral if it does not change the $\mathrm{dcl}$ relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non-existence of neutral sets in various models of ${\sf PA}$. We show that cofinal extensions of prime models are neutrally expandable, and $\omega_1$-like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first-order property. In the last section, we study a local version of neutral expandability.