Theorem 1.1.4. Let I be a nonempty set of integers that is closed under addition and subtraction. Then I either consists of zero alone or else contains a smallest positive element, in which case I consists of all multiples of its smallest positive element.

What exactly does it say? Is this the well-ordering theorem?

• It uses the fact that the usual ordering of $\mathbb{N}$ is a well-ordering, as does every induction argument. Sep 3 '13 at 17:49
• Plz add more tags. Sep 3 '13 at 18:41
• Please see my post: math.stackexchange.com/q/4080336/424260 Mar 28 '21 at 11:43

One interpretation, in terms of abstract algebra, is that every subgroup of a cyclic group is cyclic, and $I$ is a subgroup of the cyclic group of integers with the group operation of addition. The rest of the statement is just characterizing what a generator of the cyclic subgroup $I$ looks like, when the generator is restricted to be positive (because if $I$ is not the trivial subgroup, you will have one positive and one negative generator, which are additive inverses of each other).