  Turing actually constructed a non-decidable set. As we will see, he did this using a diagonal argument. The diagonal argument goes back to Georg Cantor who used it to show that the real numbers are uncountable. Gödel used a similar diagonal argument in his proof of the Incompleteness Theorem in which he constructed a sentence, \(J\), in number theory whose meaning could be understood to be, “\(J\) is not a theorem.”