[FOM] What can't be forced?

[Ilya Tsindlekht]

On Tue, May 22, 2007 at 07:29:37AM -0400, Robert Lubarsky wrote:
> > A more precise question to ask is "if we start with the minimal 
> > countable transitive model M, which consists of L(alpha) for the first 
> > alpha where this gives a model of ZFC, is there any statement known to 
> > be consistent which does not hold in M(G) for any generic set G"?
> 
> Here are some examples, perhaps none of which will satisfy you:
> not Con(ZF)
I think he asked about proposition which can hold in transitive models,
and I believe not Con(ZF) cannot since it implies existense of
non-standard integers (assuming ZF is actually consistent).