FOM: Real-valued measurables
Joe Shipman
shipman at savera.com
Wed Feb 23 16:26:46 EST 2000
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
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