[FOM] Eliminability of AC
Robert M. Solovay
solovay at Math.Berkeley.EDU
Tue Apr 1 15:52:58 EDT 2008
On Mon, 31 Mar 2008, James Hirschorn wrote:
>
> A basic example for absoluteness (but not exactly Schoenfield's absoluteness)
> is:
>
> Theorem. Every Delta-1-2 set of reals is Lebesgue measurable.
This is not a theorem of ZFC. An old result of Godel is that (in
L) there is a "good" Delta^1_2 well-ordering of the reals. Using this, it
is straightforward to get a Delta^1-2 set of reals which is non Lebesgue
measurable.
There is an old result of mine (well-known but alas unpublished)
that every provably Delta^1-2 set is Lebesgue measurable. Here the word
"provable" means that the equivalence between the Sigma^1-2 and Pi^1-2
definitions of the set in question is provable in ZFC.
--Bob Solovay
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