2003 seminar talk: Well-Orderings of the Reals and Real-valued Measurability
Talk held by Andres Caicedo (KGRC) at the KGRC seminar on 2003-10-07.
I will prove that (if there are measurables, then) there are extensions of V where the continuum is real-valued measurable and there is a Sigma-2-2 well-ordering of the reals. I will also show that natural strengthenings of the concept of real-valued measurability imply that the reals are not Sigma-2-n well-orderable for any n (so the result is not vacuously true) and that real-valued measurability implies that no well-ordering of the reals belongs to L(R) (so the result is non-trivial.) Depending on time, I might mention what is known with respect to real-valued measurability of the continuum and Sigma-2-1 well-orderings, which is a much more complicated story.