Model Constructions

Abstract

In this chapter we shall introduce various methods for the construction of models of an axiom system Σ of sentences in a formal language L. In Chapter 1 we have already encountered a method, in the form of the so-called term-models, for obtaining at least one model of Σ. Having presented that “absolute” construction, we shall now present a series of “relative” constructions. These relative methods allow us to start with one or more given models of Σ, and to produce a new model. The methods considered here do not (as often occurs in mathematics) depend on the particular axiom system Σ (e.g. the direct product of groups is again a group, while the analogue of this for fields does not hold); rather, our methods will work in every case. This will be guaranteed by the fact that an L-structure $\mathfrak{A}'$ obtained by these methods from an L-structure $\mathfrak{A}$ is elementarily equivalent to $\mathfrak{A}$; i.e.

$$ \textit{ every $L$-sentence $\varphi$ that holds in $\mathfrak{A}$ also holds in $\mathfrak{A}'$, and conversely.}$$

Therefore, if $\mathfrak{A}$ is a model of Σ, then so is $\mathfrak{A}'$, independent of which axiom system Σ⊆Sent(L) we are working with.