Models of expansions of $\mathbb N$ with no end extensions

by Shelah. [Sh:937]
Math Logic Quarterly, 2011
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 N such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of N such that expanding N by any uncountably many of them suffice. Also we find arithmetically closed A with no definably closed ultrafilter on it

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