Cofinal Extensions and Coded Sets

TL;DR

This paper characterizes the sets of subsets of a cut in a model of Peano Arithmetic that can be coded by cofinal extensions, detailing conditions for various types of extensions such as non-filling, filling, and n-filling.

Contribution

It provides a comprehensive characterization of coded sets in cofinal extensions of models of Peano Arithmetic, including conditions for different extension types.

Findings

Identifies sets of subsets of I that can be coded by cofinal extensions.

Provides criteria for non-filling, filling, and n-filling extensions.

Enhances understanding of the structure of models of Peano Arithmetic.

Abstract

Let M be a model of Peano Arithmetic that is countably generated over an exponentially closed cut I. We characterize those sets X of subsets of I for which there is a finitely (or countably) generated cofinal extension N of M such that I is the Greatest Common Initial Segment of M and N and X is the set of subsets of I coded by the extension. We also characterize such X for which the extension N can be, in addition, one of the following: non-filling; filling; n-filling.