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Models of expansions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb N}$\end{document} with no end extensions
Author(s) -
Shelah Saharon
Publication year - 2011
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200910129
Subject(s) - uncountable set , mathematics , combinatorics , countable set
We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb N}$\end{document} such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb N}$\end{document} such that expanding \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb N}$\end{document} by any uncountably many of them suffice. Also we find arithmetically closed \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal A}$\end{document} with no ultrafilter on it with suitable definability demand (related to being Ramsey). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim