[FOM] Query on "Solovay's Inacessible"

[Robert Solovay]

One small additional remark. It follows from the work of Shelah and
myself that the following two theories have the same arithmetical
consequences:

1) ZFC + "There is an inaccessible cardinal".

2) ZF + DC + "Every set of reals is Lebesgue measurable".

-- Bob Solovay

On Sun, Apr 28, 2013 at 5:06 AM, Joe Shipman <JoeShipman at aol.com> wrote:
> According to Solovay and Shelah,
>
> Con(ZFC + Inacc) <--> Con(ZF + DC + "All sets of reals are Lebesgue measurable")
>
> I wonder how much further this equivalence can be pushed. From (ZFC + Inacc) one can prove Con(ZF);  does the axiom system (ZF + DC + "All sets of reals are Lebesgue measurable") also prove Con(ZF), or any other arithmetical sentence that is not a consequence of ZF?
>
> -- JS
>
>
> Sent from my iPhone
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