There is a wide variety of statements $S$ for which forcing can be used to prove the (relative) consistency of both $S$ and $\lnot S$ with $\mathsf{ZFC}$.

For example, Cohen developed forcing to prove the consistency of $\lnot\mathsf{CH}$, but we can also prove $\mathsf{CH}$ consistent this way: We can add to the universe a well-ordering of the reals of type $\omega_1$ without adding any reals. This, by the way, shows that any projective statement provable from $\mathsf{CH}$ is provable without it.

There are some restrictions: $V=L$ cannot be forced. This is because $L$ is *forcing invariant* (we can also say *generically absoute*): If $W$ is a forcing extension of $V$, then $L$ in the sense of $W$ is the same as $L$ in the sense of $V$. This is true of larger *core models*, and is one of their key features. It allows us to prove results about the universe by arguing about forcing extensions, and is particularly useful when proving consistency strength lower bounds (and in the development of the general theory of core models).

Similarly, if $\alpha$ is a countable ordinal, then we cannot use forcing to make $\alpha$ uncountable (the idea being that if we already see a surjection from $\omega$ onto $\alpha$, adding more sets is not going to make this go away).

Other examples of statements $S$ that can be forced either way are: $2^{\aleph_1}=\aleph_{17}$, $\mathfrak b=\omega_1<\mathfrak c$, any algebraic homomorphism between Banach algebras is continuous, etc. The list can go on forever. If we assume the presence of large cardinals in the universe, then even more examples can be given.

Joel Hamkins and his collaborators (notably, Benedikt Loewe) have worked on this, making explicit interesting connections with modal logic (think of $\diamondsuit\phi$ as "it is forceable that $\phi$" and of $\square\phi$ as "$\phi$ holds in all extensions"). See here for a starting point.

Now, an interesting fact about forcing is how ubiquitous it is. This distinguishes it from other techniques. Woodin's $\Omega$-conjecture can be informally described as stating that any $\Pi_2$ statement we can verify consistent (with large cardinals) in fact can be forced (over arbitrary models with large cardinals). (This is an informal description. A more technical presentation would perhaps obscure the point here. See this survey by Bagaria, Castells, and Larson, for details.)

One of the reasons why the $\Omega$-conjecture is of interest is that it holds in all known "$L$-like" models. It is also forcing absolute, so its consistency cannot be verified by forcing (unless we already know it is true). Another reason is that there are a few examples of $\Pi_2$ statements we do not know how to force over arbitrary models, though we can prove consistent with appropriate large cardinals (by forcing over special ground models). One is: "There is a $\mathbf{\Sigma}^2_1$-well-ordering of the reals, and the continuum is a real-valued measurable cardinal", see here.

In his book **The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal**, Hugh suggests a few statements of large cardinal character (stronger than "there is a limit $\lambda$ and a nontrivial $j:V_\lambda\to V_\lambda$") that appear to be *fragile*, meaning that it is not clear that they are preserved by small forcing. Whether these statements can be proved consistent with $\mathsf{CH}$ or with $\lnot\mathsf{CH}$ does not seem to be a matter of forcing. Something different would be needed here.

needforcing only for the consistency of $S$. $\endgroup$ – Andreas Blass Aug 11 '13 at 4:4211more comments