To flesh out my comment from your Q&A session, given a countable M, consider the theory T consisting of the atomic diagram of M, together with ZFC + V = L. You proved that this theory is (finitely) consistent in your “nonstandard” case (note that in this proof, we do not need to assume that M is omega nonstandard). So by the omitting types theorem, there is a model N of T such that N is an end extension of M.

]]>So in my treatment of this issue in the book I am currently writing on the philosophy of mathematics, I take a somewhat softer tone, while making essentially the same points.

I find these issues quite subtle and interesting, for both mathematical and philosophical reasons.

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