TITLE: Does proof by contradiction assume that math is consistent?
QUESTION [25 upvotes]: The standard proof by contradiction goes like

It is known that $P$ is true.
Assume that $Q$ is true.
Using the laws of logic, deduce that $P$ is false.
Rejecting this contradiction, we are forced to accept the falsity of $Q$.

In rejecting the contradiction we implicitly assume that mathematics is consistent. However, doesn't Godel's (Second) Incompleteness Theorem tell us that the consistency of mathematics cannot be proven? Does this pose a problem?

REPLY [9 votes]: If logic is consistent, we have proven Q false.
If logic is inconsistent, then all statements are false (and true, simultaneously).
Either way, Q is false.