more introductory introduction
These two sentences are currently in the introduction: "More formally, the integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition. Like the natural numbers, the integers form a countably infinite set." Due to all the undefined terms, which are much deeper mathematically than the integers themselves, I think this pair of sentences is a strong turn-off to anyone trying to read the article (except someone who already knows all the content in the article, but we're nor writing for them).
What is the typical reader of this article likely to want (need) to know about integers? Things like the difference between "integer" and "whole number" and "natural number", probably. Ways in which integers are important - in divisibility/factorization, perhaps. I doubt they're looking for a formal construction ... but if we decide to include one (or more), then let's start by pointing out to the reader the probably surprising fact that mathematicians deem a formal construction even necessary! Similarly, if we're going to compare and contrast the integers with more general objects (groups/rings/number fields), let's set the stage for why that's important in mathematics - rather than just jumping in with something traumatizing like "The integers form the simplest example of a ring of integral elements over a number field, albeit a special one because its class number is 1."
In short (and I am a cheerful broken record on this point), let's try to make this article, and all mathematics articles, read like a narrative that's accessible for the majority of its length to a generally intelligent non-scientist. - Greg Martin 23:42, 11 May 2007 (CDT)