The Unapologetic Mathematician

Mathematics for the interested outsider

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: (a,b)(c,d)=(ac+bd,ad+bc)=(0,0). Since each of a, b, c, and d is a natural number, the order structure of \mathbb{N} says that for ac+bd=0 we must have either a or c be zero and either b or d as well. Similarly, either a or d and either b or c must be zero. If a is not zero then this means both c and d, making (c,d)=0. If b is not zero again both c and d are zero. If both a and b are zero, then (a,b)=0. 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 \mathbb{Z} 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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April 3, 2007 - Posted by | Fundamentals, Numbers

2 Comments »

  1. [...] uniqueness of the integers It’s actually not too difficult to see that the integers are the only ordered integral domain with unit whose non-negative elements are well-ordered. So let’s go ahead and do [...]

    Pingback by The uniqueness of the integers « The Unapologetic Mathematician | April 4, 2007 | Reply

  2. I’m not quite sure I agree with your argument for Z to be an integral domain, if (ac+bd, ad+bc)= (0,0) that only says that ac+bd=ad+bc, you cannot conclude that the components are equal because what was meant by the = was an equivalence relation

    Comment by Sharkasm | December 24, 2010 | Reply


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