The definition of this set is contradictory, but not for the reason you've suggested. Rather, it is impossible to construct the set of numbers which can be defined by a finite string of symbols, due to Richard's paradox:
Hence, it is also impossible to construct the complement of this set. In fact, Richard's "diagonal argument" chooses precisely the sort of number we ought not to be able to choose, according to the blog post (but again, this just shows the set is not well-defined).
http://en.wikipedia.org/wiki/Richard's_paradox
Hence, it is also impossible to construct the complement of this set. In fact, Richard's "diagonal argument" chooses precisely the sort of number we ought not to be able to choose, according to the blog post (but again, this just shows the set is not well-defined).