Minimal elementary end extensions

TL;DR

The paper characterizes when a model of PA has a minimal elementary end extension coding the same subsets, based on a definability condition involving -definable sets.

Contribution

It provides a necessary and sufficient condition for the existence of minimal elementary end extensions with specific coding properties.

Findings

Characterizes minimal elementary end extensions in terms of definability.

Establishes a condition involving  and  definability for such extensions.

Connects model-theoretic extensions with descriptive set-theoretic definability.

Abstract

Suppose that ${\mathcal M}$ is a model of PA and ${\mathcal N}$ is a countably generated elementary end extension of ${\mathcal M}$. Let ${\mathfrak X}$ be the set of subsets of M that are coded by ${\mathcal N}$. Then ${\mathcal M}$ has a minimal elementary end extension that codes exactly the same subsets of M that ${\mathcal N}$ does iff every set that is $\Pi_1^0$-definable in $({\mathcal M},{\mathfrak X})$ is the union of countably many sets that are $\Sigma_1^0$-definable.