Suppose I want to prove some statement S is independent of ZFC.

Now instead of the usual approach of making models, I do the following:

- Take two large cardinal axioms L1 and L2

- Prove that ZFC + L1 $\implies$ S is true

- Prove that ZFC + L2 $\implies$ S is false

Then I argue that "Since ZFC + L1 implies S is true, then this means ZFC cannot prove S is false. Similarly, I have shown that ZFC cannot prove that S is true. Hence, S is independent of ZFC."

My questions are:

1) Does this approach work ? If so, has it actually been used ?

2) If not, what is the problem ? Is it the case that proving "S is consistent with ZFC" different from proving that "ZFC cannot prove S is false" ?

3) Or is the worry that I am adding axioms which might themselves be inconsistent with ZFC ? I realize that L1 and L2 could have been any two axioms, but I deliberately mentioned large cardinal axioms as they are generally believed to be consistent with ZFC.