"Godel constructed statements S that are neither true nor false" No, that's not what he did. He constructed a statement that is true, but that can't be proved

"Godel constructed statements S that are neither true nor false" No, that's not what he did. He constructed a statement that is true, but that can't be proved in a certain logical system. The statement was basically "This statement is unprovable with these axioms", which can either be proved - meaning you've proved something false, or can't be proved, meaning the statement is true, but is unprovable. Note that this uses two notions of "true" - provable (can be derived from axioms), and actually true.