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I have problems understanding Gödel's incompleteness theorem. I presume I have a misunderstanding of some phrase or I have to look closer at the meaning of some detail.

Gödel's second incompleteness theorem states that in a system which is free of contradictions, this absence of contradictions is neither provable nor refutable.

If we would find a contradiction, then we would have refuted the absence of contradictions. Gödel's theorem states that this is impossible. So we will never encounter a contradiction. Doesn't that mean that no contradiction exists? (If one existed, we could encounter it.) So this seems to be a proof that no contradiction exists. Thus, we proved the absence of contradictions, which contradicts the second incompleteness theorem.

This is a contradiction which I can't solve.

Daniel S.
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    Note that the theorem statement is "if a system is free of contradictions...", and thus it says absolutely nothing about the case when a system has contradictions. – Wojowu Jan 13 '15 at 10:16
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    In fact, I do not understand the concept of 'system' in this context and thus I can't tell when I'm inside and when outside of a system. – Daniel S. Jan 13 '15 at 10:42
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    A "system" referenced in the theorem is so called formal theory, which is just a collection of axioms and deduction rules which allows us to derive theorems. – Wojowu Jan 13 '15 at 11:03
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    The the question in the title, no. It does not contradict itself. – Asaf Karagila Aug 11 '15 at 07:11
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    You might want to take a look at the book "Gödel's proof" by Nagel & Newman. It may not be perfect, but it does go a long way in putting Gödel's work into context. Particularly the inside/outside the system dichotomy is well illustrated, which is central to understanding the proof. – Fasermaler Aug 25 '15 at 12:34
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    By now I'm almost certain that I confused inside/outside in the way that I prove that the system is free of contradictions, but I do this outside the system (which is actually trivial, because outside of the system, we know by definition that it's consistent). This is not a contradiction to Gödel's theorem, which only says something about the provability inside the system. – Daniel S. Sep 03 '15 at 16:58

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I know this is late, but your comment about "inside" and "outside" is the crux. Basically "proof" is always relative to the proof system.

Define a formal system as useful iff it has a proof verifier program. Godel effectively showed that there is a program that, given the proof verifier for any useful formal system $S$, will always output a sentence $Con(S)$ over PA, such that ( $\mathbb{N} \vDash Con(S)$ ) iff S is consistent. Note that this can only be stated and proven in a meta-system that is strong enough to effectively reason about programs and halting behaviour, which includes 'knowing' $\mathbb{N}$ as some 'structure' that satisfies PA. Godel-Rosser's theorem is that if $S$ is a consistent useful formal system that interprets arithmetic, then $S$ does not prove the interpretation of $Con(S)$. See this post about the specific case where $S$ is an extension of PA, and be careful not the make the same mistake as Robert Israel.

For more details and a bit on provability logic, see this post. The incompleteness theorem is actually far more general than most textbooks set out, because it applies to any useful formal system as I've defined above, including non-classical logics, yet-to-be-discovered logics, ... For the general case we must define what "interprets arithmetic" means, which I do in this post before proving the general incompleteness theorem using a different proof from Rosser's. And for a more precise calibration of what is "strong enough" for the meta-system, see this post.

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user21820
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It does not state that the absence of contradictions is not refutable. However "Gödel's second incompleteness theorem states that in a system which is free of contradictions, this absence of contradictions is not provable" is accurate.

It may be the case that ZFC for example is consistent and also that "ZFC is inconsistent" is a theorem of ZFC.

Dan Brumleve
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  • It's already covered by the definition of the system: Refuting the absence is the same as proving the existence. But the system is free of contradictions by definition, so it is impossible to prove the existence of a contradiction. As we said, that's the same as when it's impossible to refute the absence, i.e. the absence is not refutable. – Daniel S. Sep 03 '15 at 16:22
  • "But the system is free of contradictions by definition, so it is impossible to prove the existence of a contradiction." Try to distinguish "the system proves a contradiction" from "the system proves that the system proves a contradiction". The former is ruled out by the assumption that the system is consistent, but not the latter. – Dan Brumleve Sep 03 '15 at 21:24
  • Also consider adding to a consistent theory a new axiom claiming the inconsistency of the original theory. The resulting theory is also consistent, since the original theory cannot contradict the new axiom. Furthermore the resulting theory proves its own inconsistency since it assumes the inconsistency of a subtheory. – Dan Brumleve Sep 03 '15 at 21:26
  • I have the feeling there is still a part which I don't understand. Here I would think that then again, the resulting theory is not free of contradictions, and then the theorem doesn't say anything about the resulting theory, because it gives a statement only about theories which are consistent. – Daniel S. Sep 04 '15 at 12:51