The double-incompleteness theorem
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Let T be a strong enough theory, and M - its metatheory, both are consistent. Then there is a closed arithmetical formula H that is undecidable in T, but one cannot prove in M neither that H is T-unprovable, nor that H is T-unrefutable. For English translation and proof, see K. Podnieks What is mathematics: Godel's theorem and around.