Recursively saturated, resplendent models, and saturated models.

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Omitting theories of undefinable sets

Suppose M is countable recursively saturated and X is an undefinable subset of M. Is there a countable recursively saturated N such that N is an elementary end extension of M, and if YM is coded in N, then (M,Y)(M,X)?


The answer if `yes' is either (M,X)PA or Th(M,X)SSy(M).

This problem is listed in Kossak, Roman; Schmerl, James H. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006. xiv+311, but, unfortunately, with many typos.