Martin’s Maximum and definability in H(ℵ2)

In [P. Larson, Martin’s Maximum and the Pmax axiom (∗), Ann. Pure App. Logic 106 (1–3) (2000) 135–149], we modified a coding device from [W.H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co, Berlin, 1999] and the consistency proof of Martin’s Maximum from [M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annal. Math. 127 (1988) 1–47] to show that from a supercompact limit of supercompact cardinals one could force Martin’s Maximum to hold while the Pmax axiom (∗) fails. Here we modify that argument to prove a stronger fact, that Martin’s Maximum is consistent with the existence of a wellordering of the reals definable in H(ℵ2) without parameters, from the same large cardinal hypothesis. In doing so we give a much simpler proof of the original result.