The double-incompleteness theorem

Abstract

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.