Undefinability and Absolute Undefinability in Arithmetic

TL;DR

This paper surveys results on definability and undefinability in models of arithmetic, highlighting differences between standard and nonstandard models and emphasizing automorphic images in countable resplendent models of Peano Arithmetic.

Contribution

It presents a comprehensive overview of undefinability results and introduces new insights into automorphic images in nonstandard models of arithmetic.

Findings

Undefinability results differ significantly between standard and nonstandard models.

Automorphic images of subsets play a key role in understanding model expansions.

Countable resplendent models of Peano Arithmetic exhibit rich automorphism structures.

Abstract

This is a survey of results on definability and undefinability in models of arithmetic. The goal is to present a stark difference between undefinability results in the standard model and much stronger versions about expansions of nonstandard models. The key role is played by counting the number of automorphic images of subsets of countable resplendent models of Peano Arithmetic.