First-order theories

Abstract

We continue our study of Model Theory. This is the branch of logic concerned with the interplay between sentences of a formal language and mathematical structures. Primarily, Model Theory studies the relationship between a set of first-order sentences T and the class Mod(T) of structures that model T. Basic results of Model Theory were proved in the previous chapter. For example, it was shown that, in first-order logic, every model has a theory and every theory has a model. Put another way, T is consistent if and only if Mod(T) is nonempty. As a consequence of this, we proved the Completeness theorem. This theorem states that T ├ φ if and onlyif M ╞ φ for each M in Mod(T). So to study a theory T, we can avoid the concept of ├ and the methods of deduction introduced in Chapter 3, and instead work with the concept of ╞ and analyze the class Mod(T). More generally, we can go back and forth between the notions on the left side of the following table and their counterparts on the right. Progress in mathematics is often the result of having two or more points of view that are shown to be equivalent. A prime example is the relationship between the algebra of equations and the geometry of the graphs defined by the equations. Combining these two points of view yield concepts and results that would not be possible in either geometry or algebra alone. The Completeness theorem equates the two points of view exemplified in the above table. Model Theory exploits the relationship between these two points of view to investigate mathematical structures. First-order theories serve as our objects of study in this chapter. A first-order theory may be viewed as a consistent set of sentences T or as an elementary class of structures Mod(T). We shall present examples of theories and consider properties that the theories mayor may not possess such as completeness, categoricity, quantifier-elimination, and model-completeness. The properties that a theory possesses shed light on the structures that model the theory. We analyze examples of first-order structures including linear orders, vector spaces, the random graph, and the complex numbers.