On large externally definable sets in NIP

On large externally definable sets in NIP

We study cofinal systems of finite subsets of \(\omega_1\). We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory, any uncountable externally definable set contains an infin...

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Veröffentlicht in:arXiv.org 2023-07
Hauptverfasser: Bays, Martin, Ben-Neria, Omer, Kaplan, Itay, Simon, Pierre
Format: Artikel
Sprache:eng
Schlagworte:
Mathematics - Logic
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Zusammenfassung:We study cofinal systems of finite subsets of \(\omega_1\). We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.
ISSN:2331-8422
DOI:10.48550/arxiv.2205.11792