> There is no Turing Machine capable of proving that this program won't halt Yes, there is. There's a machine which does nothing but print that exact proof. Y

> There is no Turing Machine capable of proving that this program won't halt Yes, there is. There's a machine which does nothing but print that exact proof. You seem to have conflated solving an instance of a problem with solving the entire problem. Turing's theorem states that no one Turing machine will get the answer right for every instance of the halting problem, but for any instance that one specific machine gets wrong, there's another machine that also gets that instance right (in addition to everything else the original machine gets right).