Real-valued Measurable Cardinals and Well-orderings of the Reals

Abstract

We show that the existence of atomlessly measurable cardinals is incompatible with the existence of well-orderings of the reals in L(ℝ), but consistent with the existence of well-orderings of the reals that are third-order definable in the language of arithmetic. Specifically, we provide a general argument that, starting from a measurable cardinal, produces a forcing extension where c is real-valued measurable and there is a Δ 2 2 -well-ordering of ℝ. A variation of this idea, due to Woodin, gives Σ 1 2 -well-orderings when applied to L[μ] or, more generally, Σ 1 2 (Hom∞) if applied to nice inner models, provided enough large cardinals exist in V. We announce a recent result of Woodin indicating how to transform this variation into a proof from large cardinals of the Ω-consistency of real-valued measurability of c together with the existence of Σ 1 2 -definable well-orderings of ℝ. It follows that if the Ω-conjecture is true, and large cardinals are granted, then this statement can always be forced.