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JoshuaZ comments on No one knows what Peano arithmetic doesn't know - Less Wrong
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Zetetic gave a good response. I'd just like to note that the title is slightly inaccurate. Consider for example Goodstein's theorem. PA can't know whether Goodstein's theorem is true or not, but based on its truth in very weak, reasonably intuitive extensions of PA we should probably believe its truth. So we can know something that PA doesn't know albeit in a weak sense.
The title is a reference to No One Knows What Science Doesn't Know, from which I infer that it's supposed to be read as "There is some set {what PA knows}. No one can know exactly where the boundaries of this set are.", not "There is some set {what PA knows}. No one can know the truth value of any proposition not in this set.".