The double-incompleteness theorem
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 an English translation, see Section 6.2 of my book "What is Mathematics? Godel's Theorem and Around."