
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 is a model of PA and is a countably generated elementary end extension of . Let be the set of subsets of M that are coded by . Then has a minimal elementary end extension that codes exactly the same subsets of M that does iff every set that is -definable in is the union of countably many sets that are -definable.
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