[FOM] Query on "smallness" bounds on 2^{\aleph_0} with all sets measurable

[Monroe Eskew]

Choice proves the existence of a non-Lebesgue measurable set, so that
statement is inconsistent with ZFC.  If however there is some
countably complete measure on R (not translation invariant) then the
continuum is real-valued measurable, and weakly Mahlo.  For some
details see Kanamori 03, and
http://en.wikipedia.org/wiki/Measurable_cardinal.

On Fri, Mar 20, 2009 at 6:12 PM,  <todunion at ucollege.edu> wrote:
> Is anyone aware of a known minimum value (or large cardinal property)
> for how small the continuum could possibly be yet still have "all
> sets Lebesgue measurable"?  (I think {\aleph_1} is automatically
> ruled out, but just to be on the safe side, let me add: assuming ZFC
> and not-CH)
>
> I know there are results of Solovay linking consistency of measurable
> cardinals with existence of a (non translation invariant of course)
> extension of Lebesgue measure to all subsets of [0,1]; and that
> (axiom):"2^{\aleph_0} is RVM" is equiconsistent with such an extension.
>
> Tom Dunion
> duniont_at_aol.com
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