How do we know whether a certain statement is provable or not? Certain statements are known to be unprovable within a given axiomatic system; the continuum hypothesis within ZFC is an example. We can either add the continuum hypothesis, or its negation, to ZFC, and proceed with the new consistent set of axioms.
My question is: how do we know whether a given statement (say, within ZFC) is provable or unprovable? I'm intrigued because I read that for many years people sought a proof of the continuum hypothesis within ZFC (e.g. it was no. 1 on Hilbert's list of pressing problems), which turns out to be impossible. Could it be that there are "unsolved" problems out there for which people are searching solutions, while in fact those statements are unprovable? Must a mathematician always live with the fear of doing Sisyphus' labour? As specific examples, could the Riemann hypothesis or the Goldbach conjecture actually be unprovable within ZFC?
 A: What you are interested in here is the study of Consistency Results.
Definition:
A statement $\phi$ is independent of ZFC if neither $\phi$ nor $\neg\phi$ are provable from ZFC.
Kunen's "Set Theory An Introduction to Independence Proofs" goes into great detail on how to show consistency results.
If those methods ( or any valid method) can be applied to $ \phi$ and $\neg \phi$ then $ \phi $ is independent of ZFC i.e. not provable from ZFC, and not disprovable.
A classical example:
One method is the careful construction of models. It is possible to construct a model of ZFC where the Continuum Hypothesis is true; and it is possible to construct a model where the Continuum Hypothesis is false.
You can conclude, the Continuum Hypothesis is indepedent ( not provable from) ZFC.
The method above uses Forcing, which is the basis for many consistency results. However, it is quite technical and so I reccomend you refer to Kunen for details.
Note: Of course, this is all under the assumption that ZFC is consistent. That is, for all $\phi $, ZFC cannot prove both $\phi$
and $ \neg \phi $
