It's well know that the compactness theorem has many aplication in model theory, its main shows existence of nonstandars models of aritmetical and the real numbers, and not elementary of some theories like class of fields of characteristic $0$, the class of torsion groups, the class of connected graphs ect...

I wanna to know what's your favorite application of this theorem and a short proof. My favorite aplication is the short and elegant proof of Robinson's theorem:

Robinson Consistency Theorem: https://en.wikipedia.org/wiki/Robinson%27s_joint_consistency_theorem

Proof: If $T_1 \cup T_2$ is inconsistent for compactness there exist finite $\Sigma_1 \subset T_1$ and $\Sigma_2 \subset T_2$ such that $\Sigma_1 \cup \Sigma_2$ is inconsistnet. Take $\sigma_1 =\bigwedge \Sigma_1$ and $\sigma_2 =\bigwedge \Sigma_2$ then $\{\sigma_1, \sigma_2\}$ is inconsistent, it follows that $\sigma_1 \vDash \neg \sigma_2$. By interpolation theorem there is $\theta$ such that $T_1 \cap T_2 \not\vDash \theta$ and $T_1 \cap T_2 \not\vDash \neg\theta$.


closed as primarily opinion-based by Hanul Jeon, Scientifica, Shailesh, Ethan Bolker, ancientmathematician Oct 3 '18 at 15:53

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    $\begingroup$ I mean... Almost any result of model theory or that uses model theoretic techniques is going to employ compactness at some point... $\endgroup$ – Malice Vidrine Oct 1 '18 at 23:43
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    $\begingroup$ I love the compactness theorem as much as anyone, and I appreciate that you want to celebrate it. But I'm not sure this is an appropriate post for this site. $\endgroup$ – Alex Kruckman Oct 2 '18 at 1:18