Why the proof by contradiction is a valid proof in Mathematics?

Question

In logic and mathematics proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that assuming the proposition to be false leads to a contradiction, does the proof by contradiction assume that Mathematics is consistent? What if some part of its axiomatic system is inconsistent, especially Godel's Second Incompleteness Theorem tells us that the consistency of Mathematics cannot be proven?

Answer 1

Let's assume you proved $A$ with a proof by contradiction, but you do not trust that result: You suspect that possibly there exists some tiny, tiny contradiction in Math, i.e., a statement $P$ such that both $P$ and $\neg P$ are provable. Well, if you think so, you need to believe even more that $A$ can be proven because $(P\land \neg P)\to A$ is tautological.