Wouldn't work, since 5 is not a member of the set of indescribable numbers. I am still not sure that the original example is good, since serious difficulties wi

Wouldn't work, since 5 is not a member of the set of indescribable numbers. I am still not sure that the original example is good, since serious difficulties with the axiom of choice have to do with infinite collections of sets, especially uncountably infinite, not with choosing a member of a single difficult-to-describe set. But you can't handle these non-describable numbers very easily.