1
$\begingroup$

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

$\endgroup$
1
  • 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
2
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.