[FOM] Query on "Solovay's Inacessible"
JoeShipman at aol.com
Sun Apr 28 20:33:50 EDT 2013
That's what I thought. The reason I brought this up is that a physicist has been telling me that the math for Quantum Field Theory would be easier and more natural if you scrapped AC and used the axiom that all sets of reals are Lebesgue measurable. Physicists don't care about nonmeasurable sets.
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On Apr 28, 2013, at 6:15 PM, Andres Caicedo <caicedo at diamond.boisestate.edu> wrote:
Yes, of course. The theory ZF+DC+Lebesgue measurability implies that omega_1 is inaccessible in L. This shows that the theory proves that there is a standard model of ZFC and much more.
On Sun, 28 Apr 2013, Joe Shipman 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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