Task is to find a function, which for any non-empty subset of the reals, gives an element of this subset. For this easy looking problem, something none of the ideas worked, what my IT-accustomed mind could say on the first spot. I found the problem by googling for the Axiom of Choice. My source suggested that no such function exists, but it was not very clear for me.
I can not say the smallest element, because there is no guarantee that a smallest element exist.
I can not say the element most close to zero - for example, the set of the positive reals have no such element.
Any idea based on generating some "list of reals", or generating reals on some way and then selecting the first element in the list, obviously fail on that such generation is possible only for $\aleph_0$ elements.
Another option would be to use any bijection between $2^\mathbb{N}$ and $\mathbb{R}$. Thus, only a subset of the superset of the superset of the naturals should be selected, what looks easier to me (for example, a set of naturals always have a smallest element). But things became quickly very complicated with no results.
Now I start to think that no such function exists, and the reason might have something to do that the real subsets are much more (in cardinality) than the functions which we can describe by a finite amount of symbols.
If yes, does it have an easy proof?