"There is no definable well-ordering of the real numbers."
Although many mathematicians simply believe this statement to be true, actually, it is independent of ZFC. In Goedel's constructible universe L, for example, there is a definable well-ordering of the reals, having complexity Delta^1_2 in the descriptive set-theoretic hierarchy. That is, the well-ordering is a subset of the plane RxR, and it is the projection of the complement of the projection of a Borel set (and simultaneously, the complement of another such set).
The idea that well-orders of the reals cannot in principle be described or constructed is simply not correct.