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by skhunted 779 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
1 comments
naasking 779 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 779 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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