Usually that refers to Goedel incompleteness theorem and its corollaries. It concerns however only with possibility of certain proofs in a particular class of formal systems (e.g. predicate logics of various orders).

It is beyond my humble skills to suggest if the necessary formal description of the Universe would be complex enough for the incompleteness theorem to kick in or not. But bear in mind that explaining is not the same as proving. For instance, that the God created everything is a flawless explanation (although lacking in predictive strength to be of any use in sciences).