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I am aware of threads here and here which asks something similar. However, I had something very specific to ask under the same context.

I have a very elementary question about the connection between Godel's second incompleteness thorem and the method of proof by contradiction. From my limited knowledge that I have managed to gather from books and the internet, Godel's second theorem states that "No consistent, recursively axiomatized theory that includes Peano Arithmetic can prove its own consistency."

Consistency means that an axiomatic system cannot result in some statement, and its negation to be both simultaneously provable from the axioms or in other words, the axiomatic system is free of contradictions.

Proof by contradiction, I suppose, is based on the above fact. Since the axiomatic system is consistent, we should not be able to prove that a statement is both true and false at the same time. In other words, we are basically assuming that the system is free from contradictions based on which, we have proofs by contradiction. Proof by contradiction is routinely used in many mathematical disciplines like real analysis which I presume includes the Peano Arithmetic for the construction of real numbers.

However, Godel's second theorem states that we can never prove formally, within that system, its own consistency.

My question: Are we then just "assuming" consistency of the axiomatic system and then construct proofs based on contradiction? Consistency has to be assumed because Godel's second theorem makes it clear that we can never prove that the axiomatic system is consistent. Please clarify if consistency is assumed or if it is not so, please clarify the error in this thought process.

TryingHardToBecomeAGoodPrSlvr
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  • You posted an identical question here: https://math.stackexchange.com/questions/3701956/while-using-the-method-of-proof-by-contradiction-are-we-assuming-consistency Please delete the duplicate. – heropup Jun 02 '20 at 06:40
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    I am sorry about that. They are all deleted now. – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 06:48
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    If the system is inconsistent, then everything can be proved, so if you prove something by contradiction, then you have proved it, whether the system is consistent or not. – Gerry Myerson Jun 02 '20 at 06:50
  • "Proof by contradiction" is a logical tool and we formalize it through proof systems (Natural Deduction, Hilbert-style) that are consistent. This means that the rule does not produce inconsistencies: this is exactly how the rule works. We assume $P$ and if we derive $\bot$, due to the fact that the rules are sound, we have to conclude that $P$ is not true. – Mauro ALLEGRANZA Jun 02 '20 at 07:58
  • But we use logical rules to derive theorems from axioms; if the axioms are already inconsistent, then... – Mauro ALLEGRANZA Jun 02 '20 at 08:00
  • I am repeating the comment that I made to @spaceisdarkgreen. The reason why I posted this question is because when we have a proof by contradiction, strictly speaking, what we are saying is that "the hypothesis is wrong OR the system is inconsistent." Proofs by contradiction just assumes that the hypothesis is wrong. There is always the possibility that the system is inconsistent, which is never mentioned in any proof involving proof by contradiction. As we can never prove consistency of an axiomatic system, we never know for sure as to whether hypothesis is false or if system is inconsistent. – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 08:58
  • We can prove the consistency of many "axiomatic systems": the "usual" proof calculi are consistent. – Mauro ALLEGRANZA Jun 02 '20 at 11:59
  • Doesn't Godel's second incompleteness theorem state that it is impossible to prove that an axiomatic system, that includes Peano Arithmetic, is consistent? – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 12:03
  • @TryingHardToBecomeAGoodPrSlvr No, it says it's impossible for a consistent axiomatic system that "contains PA" to prove its own consistency. – spaceisdarkgreen Jun 02 '20 at 13:44
  • Aren't they both the same? Can you please help me understand the difference? In the end, we cannot prove that we have arrived at a contradiction due to wrong hypothesis because it might as well be because the system is inconsistent and there is no way, be it within the system or not, to prove that the axiomatic system containing PA is consistent. – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 13:52
  • @TryingHardToBecomeAGoodPrSlvr PA can't prove PA is consistent, but other systems that we routinely use can. For instance ZF easily proves the consistency of PA. However, ZF is a much stronger system, and if you were 'worried' about the inconsistency of PA you should be worried about ZF as well, and hence be worried that this proof of Con(PA) is meaningful. But PA can be proven consistent in much weaker systems than ZF, even systems that are much weaker than PA in some ways, e.g. Gentzen's proof (but they also have to be stronger in at least some way cause of Godel). – spaceisdarkgreen Jun 02 '20 at 15:02
  • @TryingHardToBecomeAGoodPrSlvr Also, a system proving its own consistency is a bad philosophical metric for believing it is consistent... after all, any inconsistent system can prove its own consistency. (Assuming it can express it in some sense.) A better metric is how weak of a system we can prove the consistency in and how well we intuit that the proof must be true. – spaceisdarkgreen Jun 02 '20 at 15:02
  • @TryingHardToBecomeAGoodPrSlvr For instance, the proof calculi that Mauro mentions (which is the theory with no (non-logical) axioms at all, i.e. logic stripped of any kind of specific mathemetical content, i.e. where pbc 'lives') can be proven consistent in very weak systems and the 'mechanism' that makes them consistent is very well-understood, so we have no problem saying they are consistent. As for PA, many are persuaded either by the mechanism in the ZF proof or the Gentzen proof... almost everyone believes it, but we have to acknowledge case is weaker than the one for the proof calculus. – spaceisdarkgreen Jun 02 '20 at 15:02
  • @TryingHardToBecomeAGoodPrSlvr And I'll emphasize again per the last part of my answer that the reason this gets 'philosophical' rather than is just a point of mathematics isn't exclusively due to Godel. It is because we always prove something 'somewhere' and at the end of the day there's no way to avoid circularities... we need to have faith in our basic reasoning if nothing else. Godel just tells us it will be harder to have nice situations where weak intuitively consistent theories prove the consistency of stronger less intuitively consistent ones (and I'm not saying that isn't important!) – spaceisdarkgreen Jun 02 '20 at 15:06
  • @TryingHardToBecomeAGoodPrSlvr (Also, cause at this point, one more comment is a drop in the bucket, I should say we've been avoiding another important thing: how we express consistency statements mathematically... not all systems can even do that. For instance, it would make no sense to ask whether a proof calculus proves its own consistency. It doesn't have the apparatus for making such a statement... expressing nontrivial mathematical content requires some mathematical axioms.) – spaceisdarkgreen Jun 02 '20 at 15:12

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We do not make that assumption, per se, when we prove things in a given system. Proof by contradiction is a valid rule of inference in (classical) logic... we can use it to prove things in a consistent system and we can use it to prove things in an inconsistent system. There is no assumption here... the rules are the rules.

Of course, if you are using an inconsistent set of axioms, you will be able to, in principle, to prove every statement. That's not a very good state of affairs from a philosophical standpoint.

So we want to know our axioms are consistent, not so that we can use proof by contradiction (again, that's cart-before-horse), but so that we know we're doing something meaningful. And Godel tells us (with a few caveats) that we'll never have a proof of a strong system from a weak (i.e. "safer") system, so tells us we will basically be assuming consistency rather than proving it in a philosophically satisfying way.

(Note also that even without knowledge of Godel's theorem, we would know that we need to take some consistency on faith since we need to assume the consistency of whatever simple "system" we make our most basic arguments about how proofs fit together. The hope that Godel destroys is that within this simple system, we might prove the consistency of a strong system like ZFC, where real heavy-duty mathematics is done)

spaceisdarkgreen
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    This is pretty much what I had expected. Your comment that "we will basically be assuming consistency rather than proving it in a philosophically satisfying way." confirmed my doubts. The reason why I posted this question is because when we have a proof by contradiction, strictly speaking, what we are saying is that "the hypothesis is wrong OR the system is inconsistent." Proofs by contradiction just assumes that the hypothesis is wrong. There is always the possibility that the system is inconsistent, which is never mentioned in any proof involving proof by contradiction. – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 08:54
  • @TryingHardToBecomeAGoodPrSlvr Classical logic has been remarkably successful in modelling many aspects of reality. You can be paralyzed with doubt, fretting about the remote possibilities, or you can get on with the task of making the world a better place. Perhaps you might be comforted by the fact that the only time that an inconsistency was found in a set theory, it was detected by Russell even before it was published. It took only a few years to tweak the axioms sufficiently. Einstein's relativity theory came out at the height of this supposed "crisis". Math and science went on as before. – Dan Christensen Jun 02 '20 at 14:51
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    @DanChristensen It's not really about being paralyzed with doubt or about fretting about remote possibilities. It is about rigor. If we were to be completely rigorous about our proofs, then any proof by contradiction should have concluded with "Hence, either our hypothesis is wrong, or the axiomatic system is inconsistent." The second part of the conclusion never appears in any proof, nor in the footnotes. It would suffice if this is made clear in the introductory part of the book saying – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 14:53
  • "All proofs by contradiction, technically speaking, can possibly be a result of an inconsistent axiomatic system. With the assumption that this is always a possibility, we will typically conclude such proofs by contradiction by saying that the hypothesis is false." – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 14:53
  • @TryingHardToBecomeAGoodPrSlvr Proof by contradiction has worked for thousands of years. "Tested and found effective" as they say in TV commercials. I'm guessing you don't have much experience writing formal proofs. (Like most people, even mathematicians.) If I may suggest, you should download some software that will teach you basic method of proofs including proof by contradiction. They should become second nature to you with only a few hours of practice. – Dan Christensen Jun 02 '20 at 15:14
  • @DanChristensen Now that you mentioned about my experience with proofs, I have gone through graduate level linear algebra and undergrad real analysis, even though it was a long time back. It is quite the opposite. When I write proofs, I keep wondering why people never mention certain aspects of logic, like this one. – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 15:15
  • You seem to think that I am challenging proof by contradiction, and the big theories that have been built based on many such proofs where the axiomatic system included Peano Arithmetic. I'm am not. All I am asking is the following. We know two facts for sure. 1) We can never prove that an axiomatic system which includes Peano Arithmetic, is consistent. 2) We have a proof where we have arrived at a contradiction. So then, what is the most rigorous conclusion? Is it jut that the hypothesis is wrong? Or is it that hypothesis is wrong OR the axiomatic system is inconsistent? – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 15:19
  • It's a different thing that there is a rocket science and big theories have been built which actually work in real life, and Einstein's theory is based on this etc. None of that, to me, answers my question convincingly. – TryingHardToBecomeAGoodPrSlvr Jun 02 '20 at 15:21
  • @TryingHardToBecomeAGoodPrSlvr Formal proofs are like a different world. You should look into it. Yes, it has limited application in published proofs, but it really lays bare what is and is not a valid proof -- what you are now delving into. – Dan Christensen Jun 02 '20 at 15:22
  • @Trying you keep saying “we can never prove”... who’s “we”? This ignores the fact that we use systems to prove things about system and that what we can prove is a function of what system we’re using. – spaceisdarkgreen Jun 02 '20 at 16:19
  • @Trying Also, what is “wrong”? The most rigorous conclusion is that we have proved the negation of the hypothesis in the system. What that “means” is not an exclusively mathematical question. If we know (or later learn) the system is inconsistent we have a different perspective on what that means. – spaceisdarkgreen Jun 02 '20 at 16:22
  • @Trying Consider for instance that it’s possible for a system to be consistent and prove its own inconsistency (yes, you read that right... and it’s actually a pretty simple corollary of Godel’s Theorem). “Right and wrong” (i.e. truth and falsity) are relative concepts too, and quite distinct from provability and refutability. – spaceisdarkgreen Jun 02 '20 at 16:27
  • @TryingHardToBecomeAGoodPrSlvr "We can never prove that an axiomatic system which includes Peano Arithmetic, is consistent." So what? No known inconsistencies have resulted from the use of proof by contradiction over thousands of years. The system works. That counts for A LOT. And if you find it distasteful, no one is forcing you to use proof by contradiction. If you want a real challenge, you can try to formulate mathematics without it. Me, I like to make use of all the tools available to me. Math is hard enough without introducing any arbitrary restrictions. – Dan Christensen Jun 03 '20 at 03:29