Properties of first-order logic

Abstract

We show that first-order logic, like propositional logic, has both completeness and compactness. We prove a countable version of these theorems in Section 4.1. We further show that these two properties have many useful consequences for first-order logic. For example, compactness implies that if a set of first-order sentences has an infinite model, then it has arbitrarily large infinite models. To fully understand completeness, compactness, and their consequences we must understand the nature of infinite numbers. In Section 4.2, we return to our discussion of infinite numbers that we left in Section 2.5. This digression allows us to properly state and prove completeness and compactness along with the Upward and Downward Löwenhiem–Skolem theorems. These are the four central theorems of first-order logic referred to in the title of Section 4.3. We discuss consequences of these theorems in Sections 4.4–4.6. These consequences include amalgamation theorems, preservation theorems, and the Beth Definability theorem. Each of the properties studied in this chapter restrict the language of first-order logic. First-order logic is, in some sense, weak. There are many concepts that cannot be expressed in this language. For example, whereas first-order logic can express “there exist n elements” for any finite n, it cannot express “there exist countably many elements.” Any sentence having a countable model necessarily has uncountable models. As we previously mentioned, this follows from compactness. In the final section of this chapter, using graphs as an illustration, we discuss the limitations of first-order logic. Ironically, the weakness of first-order logic makes it the fruitful logic that it is. The properties discussed in this chapter, and the limitations that follow from them, make possible the subject of model theory. All formulas in this chapter are first-order unless stated otherwise. Many of the properties of first-order logic, including completeness and compactness, are consequences of the following fact: Every model has a theory and every theory has a model. Recall that a set of sentences is a “theory” if it is consistent (i.e. if we cannot derive a contradiction). “Every theory has a model” means that if a set of sentences is consistent, then it is satisfiable.