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Let me begin first by saying that I have no real background in formal logic, so this question may end up being an ignorant one.

I have recently been to a talk which concerned Godel's Incompleteness Theorems. The second theorem, to my understanding, states that consistent systems cannot prove their own consistency. Therefore, I think this implies that we cannot be certain of the consistency of our system of mathematics.

If this is the case, how is the method of proof by contradiction valid? More specifically, as inconsistencies have not been eliminated completely as a possibility, could it not be that the contradiction reached is a result of the axioms of mathematics being inconsistent, rather than the initial assumption being false?

I'm not sure whether the question is formulated sufficiently clearly, but it is nonetheless a conundrum that has been puzzling me of late. Any attention would be appreciated. Thank you in advance.

I have seen a similar question asked, but I do not think the answer provided completely answers my question, as it focuses more on provability rather than consistency.

IChoi
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  • If you have $p\land q \rightarrow F$ then you conclude $\neg p\vee \neg q$. In this case either the system is not consistent and cannot prove itself consistent, it is consistent and cannot prove itself consistent, or it is not consistent and can prove itself consistent. – David P May 31 '22 at 23:25
  • I have no background in this either but I’ve often wondered the same thing. I think we simply presume that ZFC is consistent since believing otherwise is a very scary prospect... the majority of all mathematics would be suspect otherwise. We trust in its consistency, and one side effect of this is that proofs by contradiction “work”. I think. Any readers please feel free to correct me, I’d love to get a different / more rigorous perspective – FShrike May 31 '22 at 23:28
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    @FShrike: Well that's wrong. Almost all of modern mathematics can be carried out within a miniscule fragment of ZFC called BZC (Bounded Zermelo plus AC). Only set theory and model theory and higher recursion theory have significant need to use more than that. And almost all known applied mathematics can be justified within a very weak foundational system called ACA (which is far below BZC). – user21820 Jun 16 '22 at 17:45
  • (Classical) FOL is just the base logic, and has been empirically verified with absolutely no exception. So if you find an inconsistency in your foundational system, it is certainly not in FOL, and almost surely not in ACA. The proof of inconsistency will almost surely have used some axiom outside of ACA, and probably outside BZC too. – user21820 Jun 16 '22 at 17:47
  • @user21820 Interesting, thank you – FShrike Jun 16 '22 at 17:51
  • In particular, to make clear why the linked post is the real answer, note that FOL itself was invented in order to reason about boolean sentences about the real world. So it should not be surprising that FOL is 100% correct, as it is almost by fiat. Inconsistencies cannot arise if: (1) you reason about only sentences that you can justify are about the real world; (2) you do not rely on any false assumptions. For why (1) is crucial and not as trivial as it looks, take a look at Quine's paradox in this post. – user21820 Jun 16 '22 at 18:02

1 Answers1

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Let's look carefully at the situation where, in some formal system $S$, we have deduced a contradiction from an assumption $A$. There are two conclusions that people might want to reach in this situation: (1) $S$ proves "not $A$"; (2) $A$ is false.

Let's first look at (1). All the usual systems of propositional logic (not only classical logic but also intuitionistic logic) include the ability to infer "not $A$" from a proof of a contradiction under the assumption $A$. (In natural deduction systems, this ability amounts to the definition of negation. In Hilbert-style systems, this ability is more complicated, involving the deduction theorem.) So we get conclusion (1).

Now your objection is: What if $S$ is inconsistent and is "responsible" for the contradiction, while the assumption $A$ is an innocent by-stander? No problem, because an inconsistent formal system proves everything. In particular, if $S$ is inconsistent then it proves $A$. So conclusion (1) is still correct.

What about conclusion (2)? To get this conclusion (from a deduction of a contradiction using $S$ and $A$), we indeed need the consistency of $S$, as you suspected. In fact, we need more, namely that the system $S$ is sound, which means that any statement that $S$ can prove from true assumptions is itself true. Given soundness, we can use (1) and the fact that a contradiction is never true to conclude that $A$ is not true. So we get conclusion (2) if we know that $S$ is sound.

Knowing that a formal system $S$ is sound (or even the weaker fact that it's consistent) requires knowledge outside $S$ itself; that's the essence of the second incompleteness theorem. For example, I know that ZFC is sound because all its axioms are true when interpreted in the cumulative hierarchy of all sets, but that argument cannot be carried out within ZFC.

[I should acknowledge, before someone else points it out, that I have ignored paraconsistent and relevantist formal systems and also formal systems that are too weak to interpret basic arithmetic. I've also said that we need soundness to get (2), but in fact it would suffice to have partial soundness --- soundness for a class of statements that contains $A$. These points don't seem relevant to the question here, so it seems better to apologize here for ignoring them than to clutter the preceding paragraphs with enough caveats to make them true.]

Andreas Blass
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  • Thank you for your answer. I have one further question - as you say, to prove the soundness of a system, we need knowledge outside of that system. However, how can we be sure of the soundness of the system used to prove the system in question and therefore of the original system's soundness? And could this chain of logic not be repeated ad infinitum? Thanks for your help. – IChoi Jun 01 '22 at 09:14
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    @IChoi Indeed, if the only way you can be sure of something is by formally proving it in a system that you know to be sound, then you are in danger of an infinite regress of systems. As far as I can see, to avoid an infinite regress, you need, at some point, knowledge not based on formal proof. An example of that would be my intuitive knowledge about the cumulative hierarchy of sets. – Andreas Blass Jun 01 '22 at 13:59