$\Sigma=F\cup I$ where $F$ consists of sentences of $\Sigma$ having a finite model, and $I$ consists of sentences of $\Sigma$ not having a model at all. For a given $\sigma\in\Sigma$ we need to effectively decide whether $\sigma\in F$ or $\sigma\in I$ . $F\subseteq\Delta=\{\tau|\tau\mbox{ has a finite model}\}$ , and $\Delta$ is effectively enumerable by Theorem 26D. $I\subseteq K=\{\sigma|\sigma\mbox{ has no model}\}$ , and $K$ is effectively enumerable by the Enumerability Theorem (p. 142), as $\sigma\in K$ iff $\neg\sigma$ is valid. Therefore, the procedure may look as follows: we list members of both $\Delta$ and $K$ one by one and check whether $\sigma$ is in one of them (Theorem 17F, and Exercise 8 of Section 1.7).