[FOM] Does 2^{\aleph_0} = 2^{\aleph_1}?

[James Hirschorn]

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