Elementary Extensions and Symmetries

In this chapter we will see how one can learn something about a structure by using symmetries of its elementary extensions. We will examine the specific example of the ordering of the natural numbers, and we will prove that the structure \(({\mathbb {N}},<)\) is minimal. After so many pages, the reader will probably find it hard to believe that this example was my original motivation to write this book. Initially, it seemed that not much technical preparation was needed.