Abstract
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 such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of such that expanding by any uncountably many of them suffice. Also we find arithmetically closed with no ultrafilter on it with suitable definability demand (related to being Ramsey). © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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