A model in mathematic logic is called pseudo-finite, in case it satisfies only such sentences of first-order predicate logic that have a finite model. Its main part modelled based on Jouko Väänänen's article 'Pseudo- finite model theory', this text studies classic model theory restricted to pseudo-finite models. We provide a range of classic results expressed in pseudo-finite terms, while showing that a set of other well-known theorems fail when restricted to the pseudo-finite, unless modified substantially. The main finding remains that a major portion of the classic theory, including Compactness Theorem, Craig Interpolation Theorem and Lidström Theorem, holds in an analogical form in the pseudo-finite theory.
The thesis begins by introducing the basic first-order model theory with the restriction to relational formulas. This purely technically motivated limitation doesn't exclude any substantial results or methods of the first-order theory, but it simplifies many of the proofs. The introduction behind, the text moves on to present all the classic results that will later on be studied in terms of the pseudo-finite. To enable and ease this, we also provide some powerful tools, such as Ehrenfeucht-Fraïssé games.
In the main part of the thesis we define pseudo-finiteness accurately and build a pseudo-finite model theory. We begin from easily adaptable results such as Compactness and Löwenheim-Skolem Theorems and move on to trickier ones, examplified by Craig Interpolation and Beth Definability. The section culminates to a Lidström Theorem, which is easy to formulate but hard to prove in pseudo-finite terms.
The final chapter has two independent sections. The first one studies the requirements of a sentence for having a finite model, illustrates a construction of a finite model for a sentence that has one, and culminates into an exact finite model existence theorem. In the second one we define a class of models with a certain, island-like structure. We prove that the elements of this class are always pseudo-finite, and at the very end the text, we present a few examples of this class.
