[FOM] Concerning Probability Measures

Robert M. Solovay solovay at Math.Berkeley.EDU
Thu Feb 16 18:35:34 EST 2006

On Thu, 16 Feb 2006, Harvey Friedman wrote:

> I have a recollection that there is an old result of Solovay that is
> relevant to the discussion with Shipman about RVM. It is a result about ZF
> without choice (say with dependent choice):
> ZFDC + "there is a countably additive probability measure on all subsets of
> [0,1]"
> and
> ZF
> are equiconsistent (in fact, mutually interpretable).

 	Yes,this is correct [and my theorem]. It was the byproduct of my 
first attempt to prove the consistency of "All sets Lebesgue measurable" 
with DC. I think Sacks published a proof of this under the title 
"Measure-theoretic uniformity"
> Correct me if I am wrong. (Adding omega_2 random reals and taking L(R)?).

 	As I recall, even adding omega_1 random reals and taking L(R) will 
work. One has to interpret "omega-1 random reals" correctly: use the 
product measure on the product of omega_1 copies of [0,1] and then force 
with the sets of positive measure. omega_2 reals will work just as well 
and give, in fact, the same class of models.

 	--Bob Solovay
> This means that the great strength of "there is a ... " is dependent on
> having the axiom of choice.
> Harvey Friedman
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