[FOM] What can't be forced?

[Robert Lubarsky]

> 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)
large cardinal axioms
not AC -- To get the failure of Choice of various kinds, typically you take
an inner model of a forcing extension. A forcing extension of a model of
Choice will always satisfy Choice.

Bob Lubarsky