In ZFC, the existence of a well-ordering of the reals is provable, being equivalent to AC. Fine. If, in addition, one assumes V=L, one can well-order them by well-ordering the formulas. A bit harder, but OK. However, if you assume the existence of non-constructible reals, is it possible that one could prove (how, I do not have a clue) that every set which well-orders the reals is a non-constructible set, and therefore cannot be explicitly exhibited? Or, although no one has yet come up with an explicit well-ordering of the reals, is there any hope that an explicit well-ordering is theoretically possible?