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I've also posted this question on MathSE. I'm posting it here in hopes of a more comprehensive answer. The question is inspired by the following:

The applications mentioned are usually pretty complicated (except for Ax-Grothendieck, but it seems to be a rare occurrence). My question is; (also omitting these examples) are there easy "applications of model theory". For example, can we use Morley's categorecity theorem to determine a class of structures of interest "outside of model theory" has only unique models in higher cardinalities, where the actual proof of the fact is fairly complicated. I know that this works for $ACF_0$ for example, but it seems this can be established with the same amount of effort (i.e. proving Morley's theorem would be the same level of difficulty)

Also are there example of nice categorizations: For example (this may be false but something in the spirit of) "omega-stable countable groups have a nice classification as products of some class of groups".

To sum up: I'm looking for "simple" applications that follow from basic results in, say Marker's text on Model Theory for example.

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Well Steinitz's theorem on the existence of an algebraic closure for any field can be proved through an easy application of the compactness theorem for first order logic. Of course the "usual proof" is not so complicated but it requires a bit of attention, whereas with compactness it's obvious. I hope this is the kind of example you were looking for.

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  • $\begingroup$ Do you know if the uniqueness can be shown in a similar way? Maybe we could have a theory a model of which is (essentially) an isomorphism between them? $\endgroup$ – Wojowu Dec 11 '16 at 17:15
  • $\begingroup$ I haven't thought of it... I think in the end it would boil down to the usual proof, though I can't be sure. $\endgroup$ – Max Dec 11 '16 at 17:24

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