This is probably quite simple and I am just missing something. I am asked to define a choice function for the collection of all nonempty subsets of $\mathbb{Z}$ without using the axiom of choice. We have learned both Zorn's and the well ordering principle. I know how to define a choice function when the set is well ordered, but $\mathbb{Z}$ is not well ordered, so I am not sure how to approach this.
Any help would be greatly appreciated