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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?

zeraoulia rafik
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  • Proof by contradiction is not specific of mathematics: it is a general pattern of reasoning (a logical law of inference). The "real" PC (i.e. inferring $P$ from the fact that assumtpion $\lnot P$ leads to a contradiction) is specific of classical logic, where Non Contradiction and Excluded Middle hold : if a statement $\lnot P$ does not hold (because of the contradiction), then $P$ must hold. – Mauro ALLEGRANZA Oct 14 '19 at 13:41
  • That's a large overreach of the implication of Godel's SIT. There are axiomatic systems that are consistent. Plenty of non-trivial ones, too! Also, if the axiomatic system we picked is inconsistent, we are kinda doomed from the get-go. A lot of things would break, not just proof by contradiction. – Rushabh Mehta Oct 14 '19 at 13:41
  • Every proof in mathematics assumes the axioms are consistent, not just proofs by contradiction. It is the truthfulness and consistency of the axioms that makes every valid deduction from the axioms a true statement. – RyRy the Fly Guy Oct 15 '19 at 00:05

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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.

Hagen von Eitzen
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