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Will_Sawin comments on No one knows what Peano arithmetic doesn't know - Less Wrong
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More powerful formal systems are not more powerful because they know more stuff. They're more powerful because they have more confidence. PA proves that PA+Con(PA) proves Con(PA) (the proof, presumably, is not very long), but PA does not trust PA+Con(PA) to be accurate. ZFC is even more confident - has even fewer models - and so is even stronger. (Maybe it even has no models!)
That's an awesome way of putting it, thanks!
Duh, PA proves that PA+X proves X for any X, because X is one of the axioms of PA+X. Did you mean some less trivial example?
No. I meant to include a trivial example. I believed that it would help clarify the point.