I provided a direct axiomatization of integers for instance under the MO question <a href="http://mathoverflow.net/questions/23193/axiomatic-definition-of-integers/134770#134770">Axiomatic definition...</a>, in the post <a href="http://mathoverflow.net/questions/23193/axiomatic-definition-of-integers/134770#134770">Part 2: Cyclands and integers</a>. This axiomatization makes no direct reference to the natural numbers, or to any linear order.

On the other hand, a clean direct constructions of integers was mentioned by **Gerald Edgar** in a comment above--let's copy it for the sake of visual clarity:

each integer is uniquely defined as a sequence &nbsp; $(a_0\ a_1\ \ldots)$ of integers &nbsp; (called digits): $-1\ \,0\ \,1$, &nbsp; where all digits but a finite number are equal to &nbsp; $0$.

(The rest is obvious). By using only the first &nbsp; $n$ &nbsp; digits we get the continuous range of &nbsp; $3^n$ &nbsp; integers:
$$-\frac{3^n-1}2\ \ldots\ 0\ \ldots\ \frac{3^n-1}2$$

Furthermore, the ring &nbsp; $\mathbb Z[\frac 13]$ &nbsp; can be defined as all sequences &nbsp; $f:\mathbb Z\rightarrow \{-1\ \,0\ \,1\}$ &nbsp; such that all values &nbsp; $f(n)$ &nbsp; but a finite number are &nbsp; $0$.