## Axiomatic definition of integers

The real numbers can be axiomatically defined (up to isomorpism) as a Dedekind-complete ordered field.

What is a similar standard axiomatic definition of the integer numbers?

A commutative ordered ring with positive induction?

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A free commutative ring with 1 of rank 0? An initial object in the category of rings with 1? – Arturo Magidin May 1 2010 at 19:55
I don't think that this question belongs to MO. your description is right and characterizes Z. also Z is characterized by a univeral property: it is initial in the category of commutative rings. – Martin Brandenburg May 1 2010 at 19:55
I cannot answer this question, but I believe that “being the initial object in the category of rings” characterises the integers up to isomorphism; I was wondering if this property can be used somehow in order to construct a (second order?) categorical theory of integers. – Antonio E. Porreca May 1 2010 at 20:05
Isn't it considerable overkill to talk about categories? – Gerald Edgar May 1 2010 at 20:12
Here is another characterization: $\mathbb{Z}$ is a ordered ring (commutative, nontrivial, with identity) with "double induction". Every subset $S$ such that $0 \in S$ and $x \in S \Rightarrow x \pm 1 \in S$ is already $\mathbb{Z}$. – Martin Brandenburg May 1 2010 at 20:34

It's the unique commutative ordered ring whose positive elements are well-ordered.

Edit: Oh, François basically already said this, didn't notice. Should I delete this?

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It's an elegant formulation. Don't delete. – François G. Dorais May 1 2010 at 23:39
I would be very grateful for any references on proofs of the uniqueness theorem. – Victor Makarov May 2 2010 at 1:25
you should prove it by yourself. – Martin Brandenburg May 2 2010 at 9:27

The ring $\mathbb{Z}$ is the unique ordered ring which satisfies full second-order induction: $$\forall X(0 \in X \land (\forall n \geq 0)(n \in X \to n+1 \in X) \to (\forall n \geq 0)(n \in X)),$$ where $X$ varies over all subsets of $\mathbb{Z}$ (or even all sets). In the comments, Martin Brandenburg has given yet another characterization of $\mathbb{Z}$ which does not assume the ordering.

A dual characterization is that every nonempty subset of $\mathbb{Z}$ which is bounded below has a minimal element. This is closer to the characterization of $\mathbb{R}$. Note that all of these characterizations only make sense in standard second-order logic, but the proposed characterization of $\mathbb{R}$ has the same problem.

The ring of integers also has categorical characterizations. For example, as proposed in the comments, $\mathbb{Z}$ is initial object in the category of (ordered) rings. See this question for related information.

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 Having read the above comments, let me clarify that by ring I mean a not necessarily commutative ring with identity. The fact that $\mathbb{Z}$ is commutative can be seen by applying induction to the subset $X = \{x : \forall y (xy = yx)\}$ of $\mathbb{Z}$ and realizing that $X$ is closed under negation. – François G. Dorais♦ May 1 2010 at 21:03