We keep going back to the Greeks because the paradoxes of the Eleatics (such as Zeno) and the Skeptics have never been satisfactorily addressed; they apply as well to the modern formal systems you laud as to the old syllogisms. Thinking in that vein may sharpen & formalize the paradoxes in such forms as Godel's theorems, but they won't dissolve them; we need different approaches to resolving the many Skeptical arguments about, say, circularity, like this metacircular approach.