[FOM] Independence without forcing

[solovay@math.berkeley.edu]

> 
> One version of this problem is: is there a mathematically natural
> sentence phi such that
> 
> Con(ZFC) --> Con(ZFC + V=L + phi)
> 
> and
> 
> Con(ZFC)--> Con(ZFC + V=L + not-phi)
> 
> are both provable in ZFC.
> 
> (Presumably some ``unnatural" self-referential sentence phi will have
> these properties--is that true?)
> 
 
     Let R be the Rosser sentence for ZFC. Then R is airithmetic so it 
relatavizes to L [as does its negation] and Con(ZFC) implies Con(ZFC + R) and 
Con(ZFC + not R).

     Of course R would be classified as an "unnatural self-referential 
sentence".

     --Bob
Solovay