As everybody knows, the ZFC axioms may serve as a foundation for (almost) all of contemporary mathematics, and it is also well-known that several results are "indecidable" in ZFC, which means that they cannot be proved or disproved within ZFC.

It is therefore natural to look for "new axioms" to add to ZFC and make it a stronger system. But by Godel's second incompleteness theorem, the consistency of ZFC cannot be deduced from ZFC itself.

Therefore, we may add the axiom "ZFC is consistent" and obtain a new system $ ZFC_1 $ consisting of "ZFC+(ZFC is consistent)". We may iterate this, and define $ ZFC_2 $ as "$ZFC_1$+($ZFC_1$ is consistent)", etc, and we may even define $ZFC_{\omega}$, or $ZFC_{\alpha}$ for any ordinal $\alpha$.

This seems a little too easy, so my question to logicians is : is this construction completely irrelevant to logic ans set theory questions ? If so why? Is it true that the results which are classically independent of ZFC are also independent of $ZFC_1$, $ZFC_2$, $ZFC_{\omega}$ etc ?