Jouko Väänänen's Short Course on Finite Model Theory

Jouko Väänänen, who also authored the higher-order logic page at the Stanford (Internet) Encyclopedia of Philosophy, published his lecture notes as an introduction to finite model theory, based on his teaching them in Lissabon in 1993 and Helsinki in 1994.

His remarks are extremely helpful in following the basic gist. After disproving the compactness theorem (as Fagin had done in 1993), he comments:

It is only natural that the Compactness Theorem should fail, because its very idea is to generate examples of infinite models. (5)

In the lecture notes, the Trakhtenbrot Theorem of 1950 is proven, namely that the set of finitely valid first order sentences is not recursively enumerable (by mapping it onto the halting problem, which is also undecidable).  From the fact that Sat is recursively enumerable but undecidable, it follows that its "complement" cannot be recursively enumerable. This has the consequence that neither the Completeness Theorem nor the Dowardward Löwenheim-Skolem Theorem hold.