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

[joeshipman@aol.com]

>Doesn't V=L qualify? Perhaps you meant an axiom consistent with the 
negation
>of CH?
>
>James Hirschorn

Yes, I was asking for a natural model in which CH was false and 
2^{\aleph_1} was larger than the continuum.

You asked about the result that RVM implies 2^{\aleph_0} = 
2^{\aleph_1}. I don't know who first proved this, but the stronger 
result that RVM implies no cardinal smaller than the continuum has a 
power set larger than the continuum was proved by Prikry (Bulletin of 
the AMS 81 (1975), 907-909). You can find a short proof online in 
section 5E of Fremlin's monograph "Real-Valued Measurable Cardinals", 
which can be downloaded from this page:

http://www.essex.ac.uk/maths/staff/fremlin/rvmc/index.htm

-- JS



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