Cofinal Extensions and Coded Sets
James H. Schmerl

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.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Advanced Algebra and Logic
