Minimal elementary end extensions

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.

arXiv commentThis version replaces the previously withdrawn version. Not only has the spelling of the title been corrected, but so has the statement and proof of the main result