Unforceable statements in ZFC

Question

How can I determine that some particular sentence (consistent with ZFC) $\phi$ cannot be forced within a model $M$ of ZFC for no notion of forcing $P$ ? Is there a procedure that works for most of the time ? What are natural examples of such $\phi$'s ?

Answer 1

You cannot force that $V=L$.

If no large cardinals exist in an inner model, then you cannot force the existence of large cardinals. Although it is possible to "resurrect" large cardinal axioms in some cases, this usually requires them to hold in an inner model (there are some caveats here, of course, we say that $\kappa$ is "Blargh" if $V_\kappa\neq L_\kappa$ and $\kappa$ is strongly inaccessible; then starting from an inaccessible cardinal we can always force the existence of a "Blargh" cardinal; but these sort of counterexamples are artificial).

Some large cardinal axioms are upwards absolute, e.g. the existence of $0^\#$, and we know that there is no (set) forcing that adds $0^\#$ to a model. So if it does not exist, then it will not exist in a generic extension. Similarly, if $0^\#$ does exist, then it will always exist further on, and so cannot be forced to not exist.

More generally, "smallness statements" about canonical inner models, like $0^\#$ or other sharps (which can be understood as "The universe is significantly larger than the canonical inner model in question), cannot be forced into inexistence once they exist, since the canonical inner models are more-or-less-by-definition are robust under generic extensions (there are some caveats, but they are not relevant to this context). At the very least this is true below Woodin cardinals.

Any statement that will ostensibly add ordinals (e.g. change $\omega$ by adding non-standard integers; add ordinals on top of the universe; add non-standard ordinals in between the existing ordinals) will also not be forceable, since forcing does not add new ordinals.

Answer 2

Shoenfield Absoluteness Theorem says that $\Sigma^1_2$ and $\Pi^1_2$ sentences are absolute, so you cannot force them. An example is the Riemann hypotesis, which can be written as a $\Pi^1_2$ sentence, so you can't force it to be true.