The real numbers can be axiomatically defined (up to isomorphism) as a Dedekindcomplete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ring with positive induction?
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekindcomplete ordered field. What is a similar standard axiomatic definition of the integer numbers? A commutative ordered ring with positive induction? 


It's the unique commutative ordered ring whose positive elements are wellordered. Edit: Oh, François basically already said this, didn't notice. Should I delete this? 


The ring $\mathbb{Z}$ is the unique ordered ring which satisfies full secondorder 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 secondorder 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. 


Part 2: Cyclands and integers(This text is a continuation of Part 1 from the same thread). First I'll axiomatize here the cyclic groupsI'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers. CyclandsBy definition, a cycland is an ordered triple $(X\ \ 1)$ such that $X$ is an arbitrary set, symbol $$ stands for a binary operation in $X$, $1\in X$, and the following (algebraic inductionlike) axiom holds: $$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ xy\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$ IntegersDEFINITION Integers (i.e. the system of integers) is a cycland $(\mathbb Z\ \ 1)$ such that the following (additional) two axioms hold:
for arbitrary $x\in\mathbb Z$. Integers from scratchTo have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands). The system of integers is an ordered triple $(\mathbb Z\ \ 1)$ such that the following six axioms hold:
for arbitrary $x\ y\ z\ \in\ \mathbb Z$. Other standard constructions or notionsWe define in $\mathbb Z$ the usual, like:
Furthermore, in every cycland, in particular in $\mathbb Z$, there is exactly one binary operation $\cdot$ which has the following two properties:
for arbitrary $x\ y\ z\ \in\ \mathbb Z$. 


Part 1: Abelian group introduction(See Part 2 for the final answer). I feel that for a fully elegant definition of integers one needs to readdress the definition of an Abelian group. For the sake of communication I will even introduce a synonym minusop for abelian groups to stress the independence of their definition from the general groups. The general groups have an elegant definition (perhaps more than one). Then adding the commutativity axiom one gets Abelian groups. This seems to me unnecessarily complex. Let me define Abelian groups directly. Symmetric operationsSymmetric operations should not be confused with much more special commutative operations. By definition, an operation $\#:X^2\rightarrow X$ is called symmetric $\Leftarrow:\Rightarrow$ $$ \forall_{x\ y\ z\in X}\qquad x\ \#\ (y\ \#\ z)\ \ =\ \ z\ \#\ (y\ \#\ x) $$ This property will appear in the definition of minusop as an axiom. Both addition and subtraction are symmetric operations in any Abelian group. Every commutative operation is symmetric. MinusopBy definition, a minusop is an ordered pair $(X\ )$, where $X$ is an arbitrary set, and $$ is a binary operation in $X$, such that the following three axioms hold:
for arbitrary $x\ y\ z\in X$. Every Abelian group admits a standard interpretation as a minusop; and every nonempty minusop admits its standard interpretation as an Abelian group, so that Abelian groups and nonempty minusops are essentially the same objects. (I am not writing the last obvious statement in any detail to keep this post sensibly short). REMARKA class of operations even more general than symmetric is still usefulI call operation $\#:X^2\rightarrow X$ insider trading (in mathematics it's legal) $\Leftarrow:\Rightarrow$ $$\forall_{u\ w\ x\ y\ \in\ X}\qquad (u\ \#\ w)\ \#\ (x\ \#\ y)\ \ =\ \ (u\ \#\ x)\ \#\ (w\ \#\ y)$$ Every symmetric operation is an insider trading. A simplification by Emil JeřábekThe last two of the three minusop axioms above (in the previous section before REMARK) can be replaced by one, as pointed out Emil Jeřábek, as now presented below. A minusop can be axiomatized by the following two conditions:
for arbitrary $x\ y\ z\ \in\ X$. It can be seen instantly that the earlier three axioms imply the two above (the new first axiom is a repetition of the old first axiom). In the other direction, a substitution of $y$ by $x$ in axiom zero gives the old axiom 3. Furthermore, assuming the above two axioms we obtain: $$ xx\ =\ (xx)((yy)(yy))\ =\ (yy)((yy)(xx)) $$ $$ (yy)  (yy)\ =\ yy $$ which proves the old axiom 2. The new 2axiom system is equivalent to the old 3axiom system. 


The integers can also be characterized as a wellordered abelian group (Theorem 20.14 in "Modern Algebra" by Seth Warner). 

