I'm not a world expert on carbon dating, but you'd think someone writing a scientific article would know that you can keep cutting something in half over and over again without ever running out of it.
Well, in the real world, eventually you're coming down to one half of an atom or something - way past the point where you could detect the radioisotope in question.
Of course, how long it takes to get to that point depends on how much raw material you start out with. A large sample could be dated way far back.
If you're going to try and argue the inaccuracy of radiodating you'll have to explain why it matches so well with the relative ages of the fossil record as well as dendrochronology, varve dating, etc. It's highly unlikely all these dates would be wrong in exactly the same way.