Does proof by contradiction assume that math is consistent?

The standard proof by contradiction goes like

1. It is known that $P$ is true.
2. Assume that $Q$ is true.
3. Using the laws of logic, deduce that $P$ is false.
4. 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?

Solution 1:

Godel's Incompleteness Theorem does not apply to every mathematical system. However, let us suppose we are working in a system to which it applies. If our system is consistent, your proof of $\neg Q$ is meaningful. If our system is inconsistent, then everything can be proven from it. So your proof tell us that a theorem is a consequence of the axioms of our system.

Solution 2:

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.