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

  • 1
    $\begingroup$ One way to go about it is to use the fact that $\mathbb N$ is well ordered. $\endgroup$ – bradhd Mar 23 '14 at 17:48

Hint: The idea is a variant of the natural choice function on the set of non-empty subsets of $\mathbb{N}$, where we pick the least element. Well-order $\mathbb{Z}$ in some nice way. For example, let $\varphi(0)=0$, and for $z$ with absolute value $\ge 1$, let $\varphi(z)=2z$ if $z$ is positive, and $\varphi(z)=2|z|-1$ if $z$ is negative.

For each non-empty subset of $\mathbb{Z}$, choose the least element of the subset under this well-ordering.

  • $\begingroup$ oh that's perfect! I never really understood you could define your own well order, but that makes sense $\endgroup$ – user137266 Mar 23 '14 at 18:06

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