[OscarCunningham]

Since ZFC can't prove its own consistency, you can't actually prove any statement is independent (an inconsistent system proves everything). Instead what is proved is that if ZFC is consistent then the statment is independent. There's a theorem called Gödel's completeness theorem (a somewhat confusing name in the light of his more famous incompleteness theorem, but they talk about different things so there's no conflict between them) that says that a system is consistent if and only if there exists a model of that system. So the way that you normally prove "If ZFC is consistent then S is independent" is by assuming ZFC is consistent, using the Completeness Theorem to show that it has a model, altering that model to produce models of ZFC+S and ZFC+¬S, and then using the Completeness Theorem in the other direction to conclude that both ZFC+S and ZFC+¬S are consistent.