Geodesic
Fragments of Complete Extensions of PA and Mcdowell-Specker's Theorem
Publications de l'Institut Mathématique, _N_S_52 (1992) no. 66, p. 1

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We generalise Theorem 1.4 of [2] and prove that for every complete extension {\bf T} of {\bf PA} and any $n\in\omega$ there exists a model for $\Sigma_n$--fragment of {\bf T} that is not extendable (that is, a model with no proper strong elementary end-extension.) This is accomplished using a model called $\Sigma_n$-atomic. This result can be interpreted as ``McDowell--Specker's Theorem does not hold for $\Sigma_n$-fragments of {\bf PA}''.
Classification : 03C20 03C62 
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     author = {Ilijas Farah},
     title = {Fragments of {Complete} {Extensions} of {PA} and {Mcdowell-Specker's} {Theorem}},
     journal = {Publications de l'Institut Math\'ematique},
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     number = {66},
     year = {1992},
     language = {en},
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Ilijas Farah. Fragments of Complete Extensions of PA and Mcdowell-Specker's Theorem. Publications de l'Institut Mathématique, _N_S_52 (1992) no. 66, p. 1 . http://geodesic.mathdoc.fr/item/PIM_1992_N_S_52_66_a0/