[FOM] Eliminability of AC

[Robert M. Solovay]

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 	There is an old result of mine (well-known but alas unpublished) 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