 (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 the law \(x+x+\ldots +x = 0\) where the number of \(x\)\(s\) on the left hand side is the size of the model. Since the equational theory of the integers contains no such law we can tell from its theory as a whole that the integers must be an infinite set. On the other hand the rational numbers under addition and negation satisfy exactly the same equational properties as the integers, so this theory does not characterize the algebra of integers under addition and subtraction with sufficient precision to distinguish it from the rationals.