Although I still fully uphold my "Proof that every set can be well-ordered", in the face of the various objections that will be thoroughly discussed in § 2, the new proof that I give below of the same theorem may yet be of interest, since, on the one hand, it presupposes no specific theorems of set theory and, on the other, it brings out, more clearly than the first proof did, the purely formal character of the well-ordering, which has nothing at all to do with spatio-temporal arrangement.