The characterization of the integers

Okay, so we’ve seen that the integers form an ordered ring with unit, and that the non-negative elements are well-ordered. It turns out that the integers are an integral domain (thus the name).

Let’s assume we have two integers (still using the definition by pairs of natural numbers) whose product is zero: . Since each of , , , and is a natural number, the order structure of says that for we must have either or be zero and either or as well. Similarly, either or and either or must be zero. If is not zero then this means both and , making . If is not zero again both and are zero. If both and are zero, then . That is, if the product of two integers is zero, one or the other must be zero.

So the integers are an ordered integral domain with unit whose non-negative elements are well-ordered. It turns out that is the only such ring. Any two rings satisfying all these conditions are isomorphic, justifying our use of “the” integers. In fact, now we can turn around and define the integers to be any of the isomorphic rings satisfying these properties. What we’ve really been showing in all these posts is that if we have any model of the axioms of the natural numbers, we can use it to build a model of the axioms of the integers. Once we know (or assume) that some model of the natural numbers exists we know that a model of the integers exists.

Of course, just like we don’t care which model of the natural numbers we use, we don’t really care which model of the integers we use. All we care about is the axioms: those of an ordered integral domain with unit whose non-negative elements are well-ordered. Everything else we say about the integers will follow from those axioms and not from the incidentals of the pairs-of-natural-numbers construction, just like everything we say about the natural numbers follows from the Peano axioms and not from incidental properties of the Von Neumann or Zermelo or Church numeral models.

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