Any model of ZFC + GCH has a generic extension (made with a poset of size ℵ 2 ) in which the following hold: MA + 2 ℵ 0 = ℵ 2 +there exists a Δ 2 1 -well ordering of the reals. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on ω 1 . Therefore, the study of such ladders is a main concern of this article.