Martin's axiom and \(\Delta_1^2\) well-ordering of the reals (Q1354340)

Woodin showed that assuming CH and the existence of a measurable cardinal \(\kappa\) which is Woodin, there is no \(\Sigma^2_1\) well-ordering of the reals. He also showed that CH cannot be removed: Assuming \(\kappa\) is an inaccessible cardinal, there is a forcing extension, by a c.c.c. poset of size \(\kappa\), in which \(2^{\aleph_0} =\kappa\), there is a \(\Delta^2_1\) well-ordering of the reals and MA \((\sigma\)-centered) holds. The authors show that assuming \(\kappa\) is an inaccessible cardinal and the GCH holds below \(\kappa\), there is a cofinality preserving forcing extension in which \(2^{\aleph_0} =\kappa\), there is a \(\Delta^2_1\) well-ordering of the reals and MA holds. The notion of forcing is described with great care, and the exposition on the whole is outstanding.