Any finite model of the equational theory of the integers must satisfy some law the integers do not satisfy — specifically, the law x+x+...+x=0 where the number of x's equals the size of the model.

(i) It could tell us to what extent the equational laws holding of the integers characterize the integers. Since the set \(\{0, 1\}\) of integers mod 2 under addition and negation satisfies all the laws that the integers do, we immediately see that no single equational property of the integers tells us that there are infinitely many integers. On the other hand any finite model of the equational theory of the integers necessarily satisfies some law that the integers don’t satisfy, in particular t