A model of peano arithmetic with no elementary end extension

Journal of Symbolic Logic 43 (3):563-567 (1978)
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We construct a model of Peano arithmetic in an uncountable language which has no elementary end extension. This answers a question of Gaifman and contrasts with the well-known theorem of MacDowell and Specker which states that every model of Peano arithmetic in a countable language has an elementary end extension. The construction employs forcing in a nonstandard model



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Minimal elementary extensions of models of set theory and arithmetic.Ali Enayat - 1990 - Archive for Mathematical Logic 30 (3):181-192.
Models of expansions of equation image with no end extensions.Saharon Shelah - 2011 - Mathematical Logic Quarterly 57 (4):341-365.

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Models and types of Peano's arithmetic.Haim Gaifman - 1976 - Annals of Mathematical Logic 9 (3):223-306.

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