FOM: Real-valued measurables

[Joe Shipman]

Robert Solovay has answered my queries on real-valued measurables.

I had asked for a model where c is weakly inaccessible but not
real-valued measurable.  Solovay says:

>>the answer is: yes iff "ZFC + "there is an inaccessible
cardinal" is consistent.

        This follows from the following two facts [which are easy
consequences of the results in my paper on real valued measuarable
cardinals]

1) If there is a RVMC then 0# exists, and so V != L.

2) If c is weakly inaccessible, then it is an inaccessible cardinal in
L.

To go one way, if c can be weakly inaccessible, then L gives a model
of ZFC + "there is a strongly inaccessible cardinal".

To go the other way, if ZFC + "there is an inaccessible cardinal" is
consistent, drop first to L to get a model of ZFC + "there is an
inaccessible cardinal" + "0# does not exist". Now blow up the
continuum so that it is equal to kappa the first inaccessible of L.
Since 0# does not exist, c is not RVM n the resulting model.

[Alternate proof: By my results, a real valued meas. card. kappa is
*not* the first weakly inaccessible cardinal.]<<


I had also asked whether it was consistent with "ZFC+measurable" that
there was a definable real-valued measure on c.  Solovay answers
positively:

>>The following are equiconsistent:

1) ZFC + "There is a measurable cardinal".

2) ZFC + "c is a RVMC" + "Every set is ordinal definable".

[The last clause of 2) is usually express V=HOD. It easily entails that
there is a definable member of every non-empty class.]

To go from 1) to 2) make c a RVMC in the usual way, and then apply
McAloon's technique [coding above c] to make V=HOD hold.<<


-- JS