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Manfred comments on Open Thread, Jun. 8 - Jun. 14, 2015 - Less Wrong
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I think the typical example is if you do a search for a proof of inconsistency in Peano arithmetic. You don't expect to find any inconsistencies, but you can't prove that you won't.
More like trying to find the Godel statement of the universe; it provably exists, and provably cannot be positively identified.