[FOM] Does 2^{\aleph_0} = 2^{\aleph_1}?
James Hirschorn
James.Hirschorn at univie.ac.at
Tue Oct 30 15:52:24 EDT 2007
On Sunday 28 October 2007 19:18, joeshipman at aol.com wrote:
> a real-valued measurable cardinal (any real-valued measurable cardinal
> is weakly inaccessible and no larger than c, but it could be smaller).
I assume there is typo in your statement in brackets? (Perhaps you
meant "atomlessly measurable" since real-valued measurable includes
two-valued measurable according to the accepted terminology.)
>
> My main point was unaffected by this, since the model you describe is
> directly constructed to make 2^{aleph-0} be less than 2^{aleph-1},
> while I was asking if any models of ~CH which were not specifically so
> constructed had nonetheless turned out to have this property.
The model you are asking for would have 2^{aleph-1} > aleph-2.
Thus an easier question is if there exist "naturally occurring" models of
2^{aleph-1} > aleph-2. Now RVM gives a positive answer, but are there any
others? I can't think of any, but maybe there are well known examples I am
unaware of. I have the impression that with the exception of RVM, every
known "natural" axiom that has something to say about 2^{aleph-1}, implies
2^{aleph-1} = aleph-2.
James Hirschorn
More information about the FOM
mailing list