[FOM] Independence without forcing

solovay@math.berkeley.edu solovay at math.berkeley.edu
Sun Jul 13 22:06:51 EDT 2003

> 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 


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