TITLE: Proof by contradiction and Incompleteness
QUESTION [8 upvotes]: Does Gödel's incompleteness theorem imply that proofs by contradiction don't work?

REPLY [3 votes]: No.  And perhaps one should stop there.
Many theories are incomplete, because we want them to be.  Look for example at the elementary theory of groups.  It is easy to write down a sentence that says that the group has at least $47$ elements.  This sentence is neither provable (after all, there are groups with $46$ or fewer elements), nor is its negation provable.
But when we seek axiomatizations for something like elementary number theory, we would like to choose the axioms so that any sentence is either provable or refutable.  Unfortunately, it turns out that if we want our collection of axioms to be recursive, and we want all the axioms to be true in the non-negative integers, we can't do that.  Too bad, but isn't it beautiful that we can prove we can't do that?
Proofs by contradiction work everywhere, both for theories that are incomplete by design (like the elementary theory of groups) and theories like elementary number theory, where we might wish to have a complete recursive axiomatization.
After all, proof by contradiction uses only basic reasoning.  Suppose we want to show that if $p$ is true, then $q$ must be true.  We examine possible consequences of $q$ being false.  If we show that one of these consequences is that $p$ must be false, then we know that if $p$ is true, then $q$ is true.  The rules of formal logic are designed to make that sort of reasoning available at the formal level.
The Incompleteness Theorem is mathematically fascinating.  But the Theorem has also given rise to a great deal of informal speculation, often based on misreadings of the Theorem.