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by skhunted 792 days ago
What’s interesting to note is that even if a strong system could prove its own consistency then it wouldn’t tell you anything. An inconsistent system can prove its own consistency. So if a system has a proof that it is itself consistent then you still wouldn’t know if it is consistent.
1 comments
libeclipse 792 days ago
It would tell you that the system is inconsistent
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skhunted 792 days ago
It would not. If a system can prove (I know sufficiently rich systems can’t do this but suppose it could) its own consistency you still can't conclude it is consistent.

EDIT: I'm working under the hypothetical situation in which PA could prove its consistency. I know it can't but assuming that it could prove it's own consistency you still couldn't conclude that it was consistent since an inconsistent system can prove it's consistency.

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roywiggins 792 days ago
GP is (I think correctly) stating that a system that can "prove" its own consistency is definitely inconsistent. Inconsistent systems can "prove" anything; if a system can appear to prove its own consistency, it isn't.
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skhunted 792 days ago
Yes. I know this. What I'm saying is that even if a consistent system that was strong enough could prove it's consistency then it still wouldn't tell you anything. There are system that can prove their own consistency for which it is known that they are consistent.

https://en.wikipedia.org/wiki/Self-verifying_theories

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naasking 791 days ago
These are weaker than the systems Godel's theorems are referring to, as discussed in the opening paragraph. Do these systems are not "strong enough" in the sense described in this thread.
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skhunted 791 days ago
Obviously I’m aware of this. As stated several times, my original comment refers to the hypothetical situation in which PA could prove its consistency without Godel’s theorems being true/known. One would not be able to conclude anything.

The point being, having PA prove its own consistency couldn’t tell you anything of value even in the case that Godel’s theorems were not true. This is an interesting phenomenon. The only way to know a system is consistent is to know all of its theorems.

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