I think this points to the central idea Hofstadter was trying to communicate. In order to "solve" the puzzle, you have to exit what he calls the "mechanical (m) mode" and think about the puzzle on a different level, the "intelligent (i) mode." The goal was for the reader to try and solve it by realizing blindly applying axioms of the system got nowhere. Proving the puzzle is impossible requires a mode of thinking analogous to Gödel's incompleteness theorem.