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I try to prove following problem: Let $T$ be a complete theory over countable language, then $T$ has a model $\mathfrak{A}$ with cardinality $\le 2^{\aleph_0}$ such that for each $\mathfrak{B}\models T$ and $S\subset \mathfrak{B}$ there is $R\subset \mathfrak{A}$ such that $(\mathfrak{B},S)$ is elementary equivalent to $(\mathfrak{A},R)$.

By Löwenheim–Skolem theorem, we can only consider countable models of $T$ and its subsets, but I don't know how to proceed to prove it. Any hint would be appreciated.

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    $\begingroup$ You may look at the notion of resplendent structure. $\endgroup$ – Primo Petri Sep 15 '14 at 6:58
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Some hints. First use Lowenheim Skolem to show that is suffices to consider countable $(\mathfrak{B}, S)$. Then take $\mathfrak{A}$ to be countably saturated.

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  • $\begingroup$ Thanks for your hints. But I don't know the notion of saturation. There is a more elementary proofs? $\endgroup$ – Hanul Jeon Sep 15 '14 at 2:24
  • $\begingroup$ @tetori: This is almost the definition of saturation. Try constructing $\mathfrak A$ recursively. $\endgroup$ – tomasz Sep 15 '14 at 20:05

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