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.

That's fine. The point is that the x is also not in the set of indescribable numbers (because you have described it with f(x)=5), assuming f is describable.

The problem is that for every finite description of f, the x that fulfills f(x)=5 is not in the set of numbers than cannot be described by a finite set of symbols. (I assume here that there is only one solution of f(x)=5)