Logic – Consistency of ZFC and Proof by Contradiction

incompletenesslogicmeta-mathphilosophyproof-theory

I will start off by saying that I am an elementary student of mathematics and do not possess the deep and rigorous knowledge of most members of this site. Nonetheless, whilst learning how to do a proof by contradiction, I had a fascinating thought that I would like clarification on.

I realise that, by Gödel's incompleteness theorems, mathematical axioms must be either consistent or complete but not both. If we ever encounter an inconsistency in our (correct) mathematical reasoning, then that is a sign that our mathematical axioms are complete but inconsistent.

This may be a really stupid question, but I would rather ask it anyway:

If one is attempting a proof by contradiction (say, at the most advanced levels of mathematics) and they encounter a contradiction, how would we know that it is the contradiction we were searching for and NOT an inconsistency of our axioms? I realise that, given the success of our axioms thus far, this is extremely unlikely, but if the unlikely did happen in such a situation, how would we know?

Again, I apologise if this is a stupid question, but I would greatly appreciate it if someone would entertain my thought.

Best Answer

how would we know that it is the contradiction we were searching for and NOT an inconsistency of our axioms?

We don't. But either way, the conclusion holds (since, if the axioms themselves are inconsistent, then all statements are true).

By that, I mean that

  • if the axioms are consistent, then $\neg P\implies \bot$ is proof that $\neg \neg P$ is true, and from that, most will conclude that $P$ is true.
  • If the axioms are inconsistent, then, because $\bot\implies P$ is true and $\bot$ is true, $P$ must also be true.

So, in both cases, the conclusion, $P$, is true (in ZFC).

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