There are 2 popular ways. 1.Set-theoretic technique : forcing, e.g. Cohen forcing which was used in the proof of the independence of the continuum hypothesis.A

There are 2 popular ways. 1.Set-theoretic technique : forcing, e.g. Cohen forcing which was used in the proof of the independence of the continuum hypothesis.A good reference is Shelah's book on forcing. 2. Model-theoretic: construct a model for ZFC in which the negation of the statement is true and another model for which the statement is true. There may be proof-theoretic techniques based on the non-existence of proofs involving finitely many steps, but I am not aware of these.