Well, Gödel set out to prove incompleteness. The first statement that comes to mind is “there is an unprovable sentence” but that is too general. What Gödel did

Well, Gödel set out to prove incompleteness. The first statement that comes to mind is “there is an unprovable sentence” but that is too general. What Gödel did was find out that “this statement is unprovable” is “definable” in Peano’s system (once it has been encoded), and that’s it. Nothing especially contrived there. Less so for a professional logician.