[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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