Consistency of ZFC and proof by contradiction

how would we know that it is the contradiction we were searching for and NOT an inconsistency of our axioms?

We don't. But either way, the conclusion holds (since, if the axioms themselves are inconsistent, then all statements are true).

By that, I mean that

• if the axioms are consistent, then $\neg P\implies \bot$ is proof that $\neg \neg P$ is true, and from that, most will conclude that $P$ is true.
• If the axioms are inconsistent, then, because $\bot\implies P$ is true and $\bot$ is true, $P$ must also be true.

So, in both cases, the conclusion, $P$, is true (in ZFC).

The short answer is that there is no simple way to distinguish an inconsistency that comes from a proof by contradiction from an inconsistency that comes from working with inconsistent axioms.

In some cases, if the axioms were the source of the inconsistency, an examination of the proof in question might show it. But if the proof is sufficiently subtle, this analysis may not be straightforward.

In many cases the proof we are working with uses very simple axioms. In these cases, it may be clear that the axioms are consistent. This is the case, for example, with the axioms for a field or a vector space.

In other cases, such as ZFC, there is a general belief in the consistency of the system, because we have been working with it for so long. Many people have tried to construct a contradiction in ZFC, and none has been found. This is not a proof that ZFC is consistent, of course (although there are some arguments that do aim to prove the consistency).

Even if ZFC was found to be inconsistent, it would not affect the vast majority of mathematical results, which do not really rely on special features of ZFC. We could reprove these results using many sets of axioms. If an inconsistency in ZFC was found, it would be interesting for logicians, but unlikely to affect most mathematicians.

When you're doing proof by contradiction, you explicitly assume some statement $A$ and after some logical process obtain a contradiction, that is, for some statement $B$ you prove both $B$ and $\neg B$. This only proves that you assumption was false, that is, it proves $\neg A$. The contradiction in ZFC would be entirely different matter - you doesn't assume anything but the axioms of ZFC and obtain the proof of both $B$ and $\neg B$. In some sense, when you prove by contradiction, you prove that ZFC, if you add $A$ to it, is inconsistent and therefore ZFC implies $\neg A$. However, inconsistency of $ZFC+A$ doesn't say anything about inconsistency of ZFC itself.