The Completeness Axiom states:
A set $\mathcal{A}$ satisfies the Completeness Axiom if for every of its non-trivial and bounded subsets, a supremum exists in $\mathcal{A}$.
If this is the statement for the completeness axiom then doesn't this imply that the set $\mathbb{Z}$ also satisfies the axiom? (Because one can easily find a supremum of a non-trivial subset of $\mathbb{Z}$.)
But then I read somewhere that $\mathbb{R}$ is the only field that is complete and if there exists another field satisfying the axiom, then it is isomorphic to the field $\mathbb{R}$ . I can't seem to understand this clearly. Does this mean that $\mathbb{R} \cong \mathbb{Z}$? Where am I going wrong?