This question arises from the talk by Voevodsky mentioned in this recent MO question. On one of his slides, Voevodsky says that
a general formula even with one free variable describes a subset of natural numbers for which one can prove, using an argument similar to the one which is used in Goedel's proof, that there is not a single number n which can be shown to belong to this subset or not to belong to it.
And in his spoken commentary he adds that there is a formula defining
a subset about which you can prove that it is impossible to say anything about this subset, whatsoever.
I interpret this as the claim that there is an arithmetically definable set $S$ for which there is no theorem of Peano arithmetic of the form $n\in S$ or $n\not\in S$. Perhaps I am misinterpreting, but can anyone supply (informally) the definition of such a set?